No. עגבנייה is a relatively new word (tomatoes came from America, so no ancient Hebrew speakers could've known of them). It comes from the root ע-ג-ב (to court, to seduce), presumably because of its red color, and similarly to its old name in German (Liebesapfel "love apple").
The word גבינה is much more ancient, and even appears in the Bible. Its root is ג-ב-נ and there are similar words in other Semitic languages such as Arabic and Aramaic.