Below are some observations with a reference request. The question I really wanted to ask is whether the minimal irrep of $ PSL(2,p) $ can always be defined over $ \mathbb{Q}(\sqrt{p}) $. In particular, for $ p $ congruent to $ 1 $ mod $ 4 $ is $ PSL(2,p) $ always a subgroup of $$ SO\Big(\frac{p+1}{2},\mathbb{Q}(\sqrt{p})\Big) $$ and for $ p $ congruent to $ 3 $ mod $ 4 $ is $ PSL(2,p) $ always a subgroup of $$ SU\Big(\frac{p-1}{2},\mathbb{Q}(\sqrt{p})\Big) $$ where unitary matrices over $ \mathbb{Q}(\sqrt{p}) $ have all their entries in $ \mathbb{Q}(i,\sqrt{p}) $.
Certainly this holds for $ p=5 $ (the icosahedral subgroup of $ SO(3) $) and for $ p=7 $ there's a $ PSL(2,7) $ subgroup of $ SU(3) $ but I'm not quite sure if it's defined over $ \sqrt{7} $ or something slightly bigger.
Observations with a reference request: (excellent reference now given in comments by David A. Craven)
For $ p < 40 $ I have noticed that the minimal degree $ d_{min} $ of a nontrivial irrep of $ PSL(2,p) $ is $$ d_{min}=\frac{p+1}{2} $$ if $ p $ is congruent to $ 1 $ mod $ 4 $ and is $$ d_{min}=\frac{p-1}{2} $$ if $ p $ is congruent to $ 3 $ mod $ 4 $.
I'm sure this fact must be well known in the literature. Does anyone have an explanation/proof/reference for this?
The value of the characters for degree $ d_{min} $ irreps are mostly $ 0,1,-1 $s also of course $ d_{min} $ and finally either $$ \frac{1}{2} \pm \frac{\sqrt{p}}{2} $$ if $ p $ is congruent to $ 1 $ mod $ 4 $ or $$ -\frac{1}{2} \pm \frac{\sqrt{-p}}{2} $$ if $ p $ is congruent to $ 3 $ mod $ 4 $. There are always exactly two irreps of degree $ d_{min} $ and their characters are related exactly by conjugation in the corresponding quadratic extension.
I assume all these patterns hold also for $ p > 40 $.
The answer is yes for both. For details see the answer by David E Speyer to the MO cross-post
https://mathoverflow.net/questions/432148/minimal-irrep-of-mathrmpsl2-p
which in turn makes heavy use of the result
https://mathoverflow.net/questions/432407/has-anyone-seen-this-construction-of-the-weil-representation-of-mathrmsp-2k
also from David E Speyer.
The key is using the Weyl representation for $ Sp_{2k}(\mathbb{F}_p) $ for the $ k=1 $ case since there is an exceptional isomorphism $$ Sp(2,p)\cong SL(2,p) $$ and so $$ PSp(2,p)\cong PSL(2,p) $$