There is no verse in the Quran saying its preferable not to eat meat but there are indications for that in the Sunnah of the Prophet.
If we look at the Muslims, traditionally, they wouldn't eat much meat:
“Meat is not a necessity in Shari’ah, and in the old days most Muslims used to eat meat, if they were wealthy, like middle class—once a week on Friday. If they were poor—on the Eids.”
- Hamza Yusuf (The Science of Shari’ah)
he continues mentioning that the Prophet was a semi-vegetarian:
So traditionally Muslims were semi-vegetarians. The Prophet was, I mean, technically, the Prophet (SAWS) was in that category. He was not a meat-eater. Most of his meals did not have meat in them.
In the Muwatta, it is mentioned that Umar Ibn Al-Khattab said about meat:
وَحَدَّثَنِي عَنْ مَالِكٍ، عَنْ يَحْيَى بْنِ سَعِيدٍ، أَنَّ عُمَرَ بْنَ الْخَطَّابِ، قَالَ إِيَّاكُمْ وَاللَّحْمَ فَإِنَّ لَهُ ضَرَاوَةً كَضَرَاوَةِ الْخَمْرِ .
Beware of meat because it is addictive as khamr (wine)
Now, there is a hadith that indicates that eating too much meat isn't good.
While one cannot find any statements about that in the Quran, there are indications in the sunnah and history of Muslims that one shouldn't eat meat too often. There are certainly hadiths speaking about over eating and the humans eager to fill his ego.