Yes, they are related. In Arabic forming words subject, ربيع and اربعة are formed by the same letters, ie. ر، ب، ع.
Spring is called as ربيع because it's occurred at September - December, ie. Season No. 4!
[Edit]: Spring is not Season No. 4 (FawwazT has mentioned it below). Also, some Arab, which I have asked personally, says Spring itself is the Third phase. -- However, the ربيع word is really connected to أربعة. Please see my comment below.
Spring, for us North hemisphere dwellers (as well as the Arabs) is around March. The sequence is usually: Spring --> Summer --> Autumn --> Winter. Usually (in India and surrounding countries) the winter crops are called ربیع as they are harvested in spring and the summer crops are called خریف as they are harvested in autumn. So I do not accept your explanation.
Interesting! If that so, الربيع comes from الرُبْعُ (means: a quarter). And, it's still related to the أربعَةٌ word.
In Lisanul Arab book, at the ربع word, the author said about the الرابع term:
والربيع : جزء من أجزاء السنة فمن العرب من يجعله الفصل الذي يدرك فيه الثمار وهو الخريف ثم فصل الشتاء بعده ثم فصل الصيف ، وهو الوقت الذي يدعوه العامة الربيع ، ثم فصل القيظ بعده ، وهو الذي يدعوه العامة الصيف ، ومنهم من يسمي الفصل الذي تدرك فيه الثمار وهو الخريف
I am sorry for not translating the statement -- as it is too complex. Also, I live in the two season country that makes it more difficult.
We can also check the explanation further here: https://ar.m.wikipedia.org/wiki/%D8%B1%D8%A8%D9%8A%D8%B9