If you assume that Jesus was not trying to trick his disciples, Matt 24:34 would be referring to events Jesus had previously mentioned that were to be fulfilled during the generation of his disciples. Those events included the end of the age:
"And Jesus said to them, "Do you not see all these things? Assuredly, I say to you, not one stone shall be left here upon another, that shall not be thrown down." Now as He sat on the Mount of Olives, the disciples came to Him privately, saying, "Tell us, when will these things be? And what will be the sign of Your coming, and of the end of the age?"" -- Mat 24:2-3 NKJV
We know from the scriptures that some of the things he mentioned were fulfilled; for example:
"But he who endures to the end shall be saved. And this gospel of the kingdom will be preached in ALL THE WORLD as a witness TO ALL THE NATIONS, and then the end will come." -- Mat 24:13-14 NKJV
In that passage Jesus said the gospel would be preached to "all the world", and to "all nations", and then the "end" would come. In a parallel passage in Mark, he said the gospel would be preached to "every creature".
"And He said to them, "Go into ALL THE WORLD and preach the gospel TO EVERY CREATURE." -- Mar 16:15 NKJV
The fulfillments and supporting scripture:
"And it came to pass in those days, that there went out a decree from Caesar Augustus that ALL THE WORLD [= the Roman Empire] should be taxed." -- Luk 2:1 KJV
"Now to him that is of power to stablish you according to my gospel, and the preaching of Jesus Christ, according to the revelation of the mystery, which was kept secret since the world began, But now is made manifest, and by the scriptures of the prophets, according to the commandment of the everlasting God, made known TO ALL NATIONS for the obedience of faith:" -- Rom 16:25-26 KJV
"For the hope which is laid up for you in heaven, whereof ye heard before in the word of the truth of the gospel; Which is come unto you, as it is in ALL THE WORLD; and bringeth forth fruit, as it doth also in you, since the day ye heard of it, and knew the grace of God in truth:" -- Col 1:5-6 KJV
"If ye continue in the faith grounded and settled, and be not moved away from the hope of the gospel, which ye have heard, and which was preached TO EVERY CREATURE which is under heaven; whereof I Paul am made a minister;" -- Col 1:23 KJV
We know from history that the temple was completely leveled: that every stone of the temple was "thrown down", as Jesus prophesied.
We know from a combination of history and scripture that the abomination of desolation was the arrival of the Roman armies on Judaean soil. Josephus records an event where General Vitellius was confronted by the Jews when he attempted to march his armies through Judaea:
"So Vitellius prepared to make war with Aretas, having with him two legions of armed men . . . But as he was marching very busily, and leading his army through Judea, the principal men met him, and desired that he would not thus march through their land; for that the laws of their country would not permit them to overlook those images [eagles] which were brought into it, of which there were a great many in their ensigns . . . Whereupon he ordered the army to march along the great plain, while he himself, with Herod the tetrarch and his friends, went up to Jerusalem" [William Whiston, Antiquities of the Jews, "The Works of Flavius Josephus Vol 3." George Bell& Sons, 1889, Book XVIII.5.3, Mat 24:15, pp.284-85]
Jesus warned the Christians to flee when they saw the abomination of desolation standing in the "holy place":
"Therefore when you see the "abomination of desolation,' spoken of by Daniel the prophet, standing in the holy place" (whoever reads, let him understand), "then let those who are in Judea flee to the mountains." -- Mat 24:15-16 NKJV
And Jesus gave the same warning to flee when the armies of Cestius Gallus arrived on Judaean soil:
"But when you see Jerusalem surrounded by armies, then know that its desolation is near. Then let those who are in Judea flee to the mountains" -- Luk 21:20-21 NKJV
These three passages are fairly convincing evidence that the abomination of desolation was the arrival of the Roman armies.
For the record, Cestius and his armies left Jerusalem for unknown reasons, giving the Christians time to flee.