Is there a relationship between Portugal and Spain being able to discover the New World and the Moor occupation?

In a backhanded way, yes.

First of all, the Arabs and Moors were driven out of Iberia 300 years before Columbus set sail, so at that point the effects from the occupation were of a historical nature.

Columbus' family was from Genoa. In those days there were two maritime empires: Genoa and Venice. This is where the most advanced technology and culture was situated. So, you might ask why did not Venice or Genoa discover the new world?

The answer is that they did not have the energy and sense of adventure and freedom. Both Genoa and Venice were hidebound with tradition and conservatism. That is why adventurers like Columbus went to Spain: they were open to new ideas and freedom to do what you wanted. Venice had so many regulations and rules and petty bureaucrats an explorer could not even get started. In Spain and Portugal the sea was wide open and there few rules. If you had money you could do whatever you wanted to do.

Fighting off the Moors had engendered this sense of freedom and independent action in Iberia. So, indirectly the Moors could be said to have created the conditions for the Age of Exploration.

The discovery of new world was nothing related with the moors. Spain with the most advanced army of europe by far that achieved to defeat the moors in Granada using the Arquebusier and cannons (western inventions) while the navy was as equal of any western countries.

Columbus was autorished to meet the Spanish Queen in order to convince her to make an expedition for a new commercial routes to Asia after several dissapoints with the Spanish University or other european kingdoms.

The three ships used were the caravels that were invented by Portuguese for long distance trips by sea (the relation with portuguese made improvements for spanish navy). The Italians were the ones who invented an improvement of the old "galley" warships that introduced the cannon (appeared in the battle of Lepanto), a predecessor of the future heavy spanish galleon.

The idea of the moors being more advanced that christians depends in which part. In armament the moors were behind. The moors didnt have any galleys or caravels.

Maybe, in science were advanced but they only could study when there wasnt a Calipha (a very conservative leader) in power. I recommend the film "The Physician" that appears the two sides of Islam. When in Spain was peace that meant that there wasnt any Calipha, but when there was, the christians had to fight to free Spain.

The battle of Las Navas the Tolosa in 1212, was The mother of all of battles of Spanish Reconquista.

The battle of las navas de tolosa was in 1212. The biggest battle of the reconquista. Crusade (all Christian kingdoms with french voluntarees) vs Yihad (Afrikan knights, marrocan infantry, Turkish soldiers, Black afrikan tribes infantry etc...) After this battle the moorish occupation entered in a big depression losing major Andalusian cities until the final of XIII. century with The almogavars unit of Christian army that was extremely aggressive against muslims taking advantage of their weak situation. The muslims totally sunk asked for peace treaty. A treaty of nearly two centuries of peace. The spanish Christian kingdoms decided The almogavars to sent to Italy freeing them from their pressure.

Once the treaty was signed, The muslims retreated to the safe province of Granada, the last muslim fortress.

In 1484,Colon arrived to spain in the middle of Granada War.

One of the greatest intellectual and academic scandals that Western public education suffered from-(certainly when I was enrolled in Public school many years ago), was an incomplete teaching of the historical truth. It is still unknown to me as to whether such an intellectual and academic scandal was deliberate and purposeful or if it was just largely overlooked, simply due to laziness and/or a cavalier temperament.

Whatever the reason(s) were, it is scandalous to think that an entire Public educational system would disregard or even deliberately omit the near 800 year rule of the Moors in Spain and the legacies of Al-Andalus and Toledo. The fact that this Public educational system insisted on using and even browbeating the (now anachronistic sounding) term, "Dark Ages", shows that there was no interest in connecting the historical dots when trying to better understand the origins of The Age of Scholasticism and in particular, the Northern Italian Renaissance-(in other words, properly and accurately chronicling...the true origins of Early Modern Europe).

The Moorish civilization of the Middle Ages, was Europe's Beacon of Light during its so-called, "Dark Ages". When looking at its architecture alone-(i.e. The Mezquita in Cordoba and of course, The Alhambra in Granada), one sees an impressive level of artistic dedication, creativity and vision.

Where does one begin when talking about the Moors of Spain? Well, how about Horticulture, Botany, urban street planning, centralized waterway systems, stucco architecture, Libraries, Schools, Palaces and many, many other contributions to the early histories of Spain, the Iberian peninsula, Europe and really....to the history of Islam and to the history of the world.

Now while the Moors certainly had their own Luminaries, they, like other civilizations prior to them, had some help along the way; that is to say, they, like the Romans centuries earlier, were deeply influenced by the writings, in particular, the scientific, medical, mathematical and philosophical writings of the Ancient Greeks-(especially, the writings of Aristotle, Archimedes, Euclid and Hippocrates).

About 1000 years ago, at the height of Muslim Andalusian civilization, Arabic translations of Ancient Greek texts, were exported from Baghdad, as well as Cairo and traveled throughout North Africa to the city of Fes in Central Morocco and to its prestigious University. Eventually, these same Ancient Greek texts arrived in Andalusia-(namely, Cordoba....the Capital of the Iberian/Western Caliphate), then a few centuries later, arrived in the Castilian city of Toledo.

While Cordoba made impressive gains and advancements in the above mentioned fields, it was really in the city of Toledo whereby the translations of these Greek texts became an interreligious and interethnic cultural experience. Typically, in 13th century Toledo, an Arabic translation of Aristotle's "Metaphysics", was then translated into Hebrew, as well as into Castilian Spanish and then subsequently, was translated into Latin. (It probably would have been very difficult for Thomas Aquinas to write his, "Summa Theologica", without the provision and availability of translated copies, more specifically, of Arabic translated copies of Aristotle).

But, like all great civilizations, Moorish Spain came to an end in 1492 with the Fall of Granada. However, while 1492 marked the official end of Moorish Muslim Spain, it did not necessarily mark the end of their legacy and impact on the evolution of Modern Spain and in particular, Modern Christian Spain.

If we look at the historical evolution of Spanish North America alone, we see how the Spanish-(while highly ambitious in spreading the Catholic faith to the Native peoples in places, such as California and the Southwestern United States), had also brought with them, a visual imprint of their architectural and horticultural development of their famed Mission Churches and Chapels.

While many Churches and Cathedrals in Spain are Gothic in structure-(i.e. in places, such as, Seville and Santiago), the Spanish-American Mission Churches, were often designed in the Mujedar style that was commonplace in Muslim Spain, in particular, within Andalusia. Just the California Missions alone-(from San Diego to San Francisco), are Churches and chapels that evoke a more Mediterranean/Andalusian style of architecture, accompanied by its finely designed gardens. There are no appearances or remnants of the older, towering Gothic style in these Mission Churches; instead, you see the finer architectural elements of the Moors-(albeit combined, with Catholic imagery and iconography).

One could focus on the Moorish architectural and horticultural legacy in North America alone and the significant impact it had on the Spanish colonial and missionary style in states, such as California.

It had more to do with the Reconquista.

There was a social pattern that Aragon, Castile, and Portugal developed as they "reconquered" the Iberian Peninsula. This same pattern was how they colonized the new world very rapidly and very cheaply.

In Reconquista a captain would gather fighting men with the promise of shares of any wealth or land they would capture. The captain would then attack nearby Moreish lands or Caribbean islands and try to conquer them. If he succeeded, he would found a town and send a letter back to the king saying he had claimed the land for the king and offering the traditional Quito Reyal, a fifth of the loot. The rest was split among the troops the captain would become mayor / governor and the new town would become the jumping off point for the next set of expiations.

The reason this was key was the cost. The crown paid nothing to have lands conquered for it, only appointing governors from successful conquests. They could afford to conquer the new world, and this was the important point. Columbus was not the first to discover the Americas, but Spain and Portugal were the first to conquer them. The Reconquista let them succeed were the Norse, Incas and Aztecs(Mexica) had been defeated.

Also key were the advancements in navigation funded by Hendry the Navigator of Portugal.

TL; DR: Yes, there is a relationship, but not of the kind you probably think.

A lot has been written on how Columbus' journey of discovery was only made possible by the fact that he was hilariously wrong about his estimates of the size of the Earth. The man himself being Genovese, his estimate would have been based largely on Italian science which yes, was influenced by Arabic and Byzantine scholars. His knowledge of seamanship as well as the technology used in his ships also came from the same sources, and all he got from Queen Isabella was financial sponsorship.

How do Moors factor into this? While they did have significant influence on Hispanic culture, having carried on some of the legacy of the Visigothic kingdoms, probably the most important role of the Umayyad Caliphate in facilitating the discovery of the New World was keeping the Spanish Christian kingdoms occupied, fractured and at times subjugated.

In practical terms, this meant that while the Holy Roman Empire, France, England, Hungary and many others were busily re-developing Roman territories or developing previously undeveloped ones for centuries on end, Spanish kingdoms were too busy fighting for their dear life. This in turn meant that by the time they managed to firmly establish themselves by the end of the 15th century, all the choice real estate in continental Europe was taken.

I can't back this up with a primary source, but I believe the argument could be made that with continental Europe taken, the Mediterranean divided between Italians and the Islamic kingdoms and North Africa being firmly in Arab hands, the opportunity of expansion into India seemed to the Spaniards an attractive proposition, even if the chance of success was a slim one.

"Some people believe that Renaissance started in Toledo". Who are these people? I think the prevailing opinion is that Renaissance started in Italy. By the way, Columbus came from Italy too.

And notice that his first voyage started in exactly the same year when the last Moorish kingdom fell, and Jews were expelled from Iberian peninsula. (As it is mentioned in the comments Moors were expelled later).

Speaking of the knowledge of antiquity. Some part of it was continuously preserved in Europe. Other parts come from two sources: Byzantine Empire and Islamic world. It is difficult to compare the contribution of these two sources, but both were important. And I suppose Byzantine influence was larger in Italy. Some connection between the Eastern Empire and Italy always existed, but very important was the immigration of intellectuals from the Eastern Empire when Constantinople was conquered by the Turks in 1453. At this time, 39 years before Columbus' first voyage a lot of people and a lot of books moved to Europe.

But I can give some specific examples of Moors contribution to Columbus voyage, and other voyages of discovery.

Alphonsine tables were developed in 1252 in Toledo. They were made by Toledo School of translators, from Islamic sources. First printed edition 1483, second 1492. Columbus used them for navigation. Alphonsine tables were used until the early 17s century.

Research institute of Henri the Navigator was established in the first half of 15s century, and it probably employed some Moorish cartographers, and certainly used Arabian literature.