Do we have to believe that good people exists?

It appears the world has relatively few good people who (per SN 55.7) both consciously follow the five precepts & speak in praise of the five precepts.

For example, there are relatively more people who more or less instinctually follow the five precepts but they don't teach their children the five precepts and they allow their children to transgress the five precepts.

From Dhammapada:

85. Few among men are those who cross to the farther shore. The rest, the bulk of men, only run up and down the hither bank.

86. But those who act according to the perfectly taught Dhamma will cross the realm of Death, so difficult to cross

Extremely few go all the way to Nibbana.

But it's not right to say that there are no good people. There may be many good people, but they may not be trying to escape suffering.

By "good people", I mean people who try to practise the four brahmaviharas - loving kindness (metta), compassion (karuna), empathetic joy (mudita) and equanimity (upekkha). One could practise these without trying to attain Nibbana.

For example, anyone who tries to help another person is obviously showing compassion.

What about the five precepts?

A monk (a full time, professional practitioner of the Dhamma) needs to practise the five precepts as perfectly as possible, as explained in the section entitled "The Moral Foundation for Jhana" by Ven. Gunaratana and also as explained in this answer.

However, a lay follower of the Dhamma may commit themselves to the practice of the five precepts but they may not strive to practise it perfectly.

Here's one example:

The lay follower who does not practice the five precepts perfectly, will not kill humans and most animals, but may occasionally kill a mosquito.

I'm sure we can find similar examples for the other four precepts.

Do we have to believe that good people exists?

You don't have to believe anything. It's entirely up to you. Buddhism is more of a practical religion based on experience rather than on belief.

As long as one is not fully Enlightened there will remain defilements (latent tendencies) in the mind.

A great focus in Buddhism is the cultivation of goodness, helpfulness, peacefulness and patience and many other positive qualities of mind. Many Buddhist (and non-Buddhists) possess those qualities so in that sense they are good people.

From a Buddhist (meditational and experiental) point of view this is a bit different. You see, a being does not really exist. What exists are moments of experience so when you see or hear a person talk these are just moments of experience such as hearing and seeing arising and passing away constantly.

The mind is like that. It arises and passes away incessantly. This is also known as momentary death. So when talking about good people we really have to talk about wholesome moments of mind arising and passing away. Being mindful is a moment of great goodness and wholesomeness so naturally a mindful person or one who trains in Buddhist mindfulness meditation can be said to be a good person.

Hope this helps.

Yes, if wishing toward Path, good householder, as it begins with Right view, wherein this view is included:

"And how is one made pure in three ways by mental action? There is the case where a certain person is not covetous. He does not covet the belongings of others, thinking, 'O, that what belongs to others would be mine!' He bears no ill will and is not corrupt in the resolves of his heart. [He thinks,] 'May these beings be free from animosity, free from oppression, free from trouble, and may they look after themselves with ease!' He has right view and is not warped in the way he sees things: 'There is what is given, what is offered, what is sacrificed. There are fruits & results of good & bad actions. There is this world & the next world. There is mother & father. There are spontaneously reborn beings; there are brahmans & contemplatives who, faring rightly & practicing rightly, proclaim this world & the next after having directly known & realized it for themselves.' This is how one is made pure in three ways by mental action."

— AN 10.176

And as the Buddha declared in the Shorter Lion Roar Sutta, the four kind of real good people are only to be found within his Sangha.

If we may agree that good requires the skillful, then:

AN3.69:9.1: There are these three skillful roots.
What three?
Contentment, love, and understanding.

Contentment, love, and understanding are worth cultivating for their own sake without needing to assess others. Indeed, the Buddha says quite clearly that:

MN8:12.12: ‘Others will have wrong view, but here we will have right view.’

Good people acting out of contentment, love and understanding are to be found in all faiths. Good people inspire us all.

I think this world is full of good people. THE first I can think of is my parents and there are many friends. The idea of "Good" is subjective. What I call good may not be the same as what you call good. The observing five precepts is a gradual training. It can have many levels. Not killing a human is not the same as not killing a virus.

Believing that good people exist is like believing that young children will mature in wisdom and intelligence. Sometimes it's hard to watch a two year old in the middle of a tantrum and think "someday this little screaming monster will be a doctor, or a saint, or someone's loving spouse", but every small child has that potential in them.

Everyone wants to be good (or at least want to think of themselves as being good). Most people only have a vague idea of how to be good; many people are confused on the very idea of what being good means. That's what the dharma is for: to turn people's heads so they are looking in the right direction when they look for good. but clearing up that confusion is a difficult task in the best of circumstances, and most people don't have access to the best of circumstances. Give them he same forbearance one might give to a child.

There is no value in assuming people are not-good. All that will do is ruin our own practice: making us cynical and jaded; leading us towards unhealthy egotistical states where we aggrandize ourselves or our masters over and above the unenlightened riffraff. We cannot lead people with a stick, or draw them forward with bile and vinegar, so we have to develop that sweetness within ourselves that sets asides such sour thoughts.