Are there any passages in Sacred Scripture that explicitly describe apocalyptic weather patterns?

You may want to consider that season can be interpreted as feast of God...ie. you can determine what is to take place per the convocation of the Feasts of Jehovah. If people are under a strong delusion, which is an end time sign, they will not understand the Feasts... the information which Jehovah has appointed. You may also want to study the Menorah and how the lamps correspond to the feasts. These are perilous times, do not look to people for Godly answers. STUDY to show YOURSELF approved and be lead by the HOLY SPIRIT which teaches ALL things.

There are two significantly divergent ways of construing "in the end times you won't be able to tell what season it is.":

The first way is the approach of the (currently marked as) accepted answer which interprets that statement to mean: "In the end times you won't be able to tell the difference between summer, autumn, winter or spring - the seasons themselves won't be distinguishable from each other".

This is not the only possible interpretation, nor do I think it the most likely original meaning of the phrase, which was most likely as an extension of the verse "no one knows the day or hour (of the timing of the Lord's return)" (cf. Matthew 24:36) through applying an implication of Matthew 24:19 quoted in the OP above particularly:

...Pray that your flight will not take place in winter...

i.e. the 'it' referred it to in the statement was (originally) referring to 'the day of the Lord' and the (particular) season of the year when the Lord will return is not yet fixed (it may be in summer, spring or autumn, but hopefully not winter if we've all been praying enough about it) except according to the Father's foreknowledge of our prayers - and He is keeping everyone else (including the pre-glorified Son) guessing.

They are referring to the Day of the Lord from

Zech 14:6 On that day there will be neither sunlight nor cold, frosty darkness.. It will be a unique day--a day known only to the LORD--with no distinction between day and night. When evening comes, there will be light. On that day living water will flow out from Jerusalem, half of it east to the Dead Sea and half of it west to the Mediterranean Sea, in summer and in winter.

I think it is possibly rooted in the corresponding rise of the final antichrist at that time. In researching his attributes, you find several curiosities that have been mostly forgotten by the modern Church. Referring to him in Daniel 7:25, Daniel indicates that he will seek to adjust time or time references in some way (along with prevailing laws).

Evidently, it comes from the “3 Days of Darkness” prophecy of Padre Pio, a Roman Catholic priest. The exact quote appears to be, “You will know when that time is approaching because the seasons will change so the only way you will know the seasons is by the leaves on the trees.”

This “not knowing the seasons except by the leaves” concept is not biblical. In fact, there are verses in the Bible which actually teach the opposite. Genesis 8:22 declares, "As long as the earth endures, seedtime and harvest, cold and heat, summer and winter, day and night will never cease." Matthew 24:32-33 exclaims, "Now learn this lesson from the fig tree: As soon as its twigs get tender and its leaves come out, you know that summer is near. Even so, when you see all these things, you know that the end is near, right at the door." See also Jeremiah 33:20-25. With these Scriptures in view, the idea of the seasons being indistinguishable in the end times does not have any biblical foundation.

Isa 24:20 The earth shall reel to and fro like a drunkard, and shall be removed like a cottage; and the transgression thereof shall be heavy upon it; and it shall fall, and not rise again.

Seasons are caused by the tilt of the earth. When the Northern hemisphere is experiencing winter the southern hemisphere is experiencing summer. The middle of the earth experiances some temperature changes as the earth moves closer and further away from the sun due to the elliptical orbit but it is not as distinctive as the poles.

The earth reeling as a drunkard will certainly change how we experiance seasons. However

Gen 8:22 While the earth remaineth, seedtime and harvest, and cold and heat, and summer and winter, and day and night shall not cease.

We are promised that summer and winter are part of God's plan for the earth as long as there is an earth and will continue to happen as long as the Earth exists.

This link from open bible has many other verses on the topic of weather in the end of days.

The only passages that come to mind regarding weather patterns in the end times have to do with clouds:

Joel 2:2 A day of darkness and of gloominess, a day of clouds and of thick darkness, as the morning spread upon the mountains: a great people and a strong; there hath not been ever the like, neither shall be any more after it, even to the years of many generations.

Zep 1:14 The great day of the LORD is near, it is near, and hasteth greatly, even the voice of the day of the LORD: the mighty man shall cry there bitterly. Zep 1:15 That day is a day of wrath, a day of trouble and distress, a day of wasteness and desolation, a day of darkness and gloominess, a day of clouds and thick darkness,

Is it the quantity of clouds? Will they be blacker than normal? We can expect unusual weather patterns when much of the earth's greenery is burnt ("The first angel sounded, and there followed hail and fire mingled with blood, and they were cast upon the earth: and the third part of trees was burnt up, and all green grass was burnt up" -- Rev. 8:7), the air is scorched, and smoke is everywhere. From what I've seen in my life on earth, dense smoke turns the sun red:

Joel 2:30 And I will shew wonders in the heavens and in the earth, blood, and fire, and pillars of smoke. Joel 2:31 The sun shall be turned into darkness, and the moon into blood, before the great and the terrible day of the LORD come.

The abundance of burning vegetation will surely cause crazy weather patterns.