Was Brahms Quoting Beethoven?

I was listening to the WQXR broadcast of the Vienna Philharmonic at Carnegie Hall last night. It was an all Brahms program: The Academic Festival Overture, The Haydn Variations, and the first symphony. I grabbed my trusty old Dover score and followed along for the performance of the symphony. I was not paying too close attention, but even so, something leaped out at me at the start of the second movement. The opening theme begins with a striking similarity to the first theme of Beethoven’s sixth symphony. I grabbed my Beethoven score, and sure enough, the first seven notes of both themes are the same. The Brahms is in E Major, the Beethoven is in F major, so the Brahms is naturally written a semitone lower. Both themes are given to the first violins. The rhythms and tempo are different.

Comparing the opening of Brahms' Symphony No. 1, Second Movement with the opening of Beethoven's Symphony No. 6, First Movement.
Brahms Symphony No. 1 vs Beethoven Symphony #6.

I’m wondering if this was a purposeful quote, or just a coincidence. Also makes me wonder if there are any other Easter eggs in “Beethoven’s 10th?”