Is q a quadratic residue of n?

Mathematica, 25 bytes

AtomQ@PowerMod[#,1/2,#2]&

Mathematica, being Mathematica, naturally has a builtin for calculating modulo nth roots, via PowerMod. If a solution exists the smallest feasible solution is returned, otherwise the original expression (plus a message).

To get an actual truthy/falsy output we pass the result to AtomQ, which checks whether an expression can be broken down. Integers are atomic, returning True, whilst the non-atomic PowerMod[q,1/2,n] returns False

Thanks to @MartinBüttner for golf tips and function hunting with me.

Par, 11 9 bytes

✶X[²x%)↔,

Each character uses just one byte; see here.

Explanation

✶              ## Read two numbers
X              ## Assign second to x
[              ## Map
 ²             ## Square
 x%            ## Mod x
)              ## 
↔              ## Swap
,              ## Count

Removed two bytes thanks to Jakube.

LabVIEW, 16 15 Equivalent bytes

Counted according to my meta post.