Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$? 
Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
Do there exist infinitely many real quadratic fields $F$ that are the coefficient field of a weight $2$ classical holomorphic eigenforms over $\mathbb Q$?

Of course the analogue of the first question for imaginary quadratic fields and elliptic curves the answer is yes. By studying the Hilbert surfaces, maybe one can prove conditionally on Bombieri-Lang that there are finitely many $F$ for which there are infinitely many isomorphism classes of such surfaces, but this of course is a long way off.
 A: A conjecture of Coleman asserts that only finitely many rings arise as the endomorphism ring of an abelian variety of given dimension defined over a number field of given degree. See [1] for an account of this conjecture. In your case, the relevant conjecture is denoted there by $C(1,2)$. To my knowledege, the only results on Coleman's conjecture in dimension >1 concern CM abelian varieties.
Regarding your second question, Ribet has shown that Serre's conjecture on representations of the absolute Galois group of $\mathbf{Q}$ [2, Conjecture (3.2.4)] implies that every abelian variety of $\mathrm{GL}_2$-type over $\mathbf{Q}$ is a quotient of $J_1(N)$ for some $N \geq 1$, see [3, Theorem 4.4]. Serre's conjecture is now a theorem of Khare-Wintenberger [4, Theorem 5.1] so that your second question is equivalent to the first.
[1] Bruin, Flynn, Gonzalez, Rotger, On finiteness conjectures for endomorphism algebras of abelian surfaces. Math. Proc. Camb. Philos. Soc. 141 (2006), No. 3, 383-408.
[2] Serre, Sur les représentations modulaires de degré 2 de $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$. Duke Math. J. 54 (1987), 179-230.
[3] Ribet, Abelian varieties over $\mathbf{Q}$ and modular forms. Modular curves and abelian varieties, 241–261, Progr. Math., 224, Birkhäuser, 2004.
[4] Khare, Serre's conjecture and its consequences, Japan. J. Math. 5 (2010), 103–125.
