3n^2+1 = x^2 only if x = 3m+1 for some m
Proof:
x hase exactly one of three forms: 3m, 3m+1 or 3m+2.

Case 3m.
x = 3m => 3m = 3r+1 for r=1,2.. which is impossible. So the statement is true for x=3m.

Case 3m+2
3n^2+1 = 9m^2+12m+4 or n^2 = 3m^2+4m => m | n. So 
n = km and n^2 = k^2m^2 = 3m^2+4m. Dividing out m we have
k^2m = 3m+4 => m | 4 or m=4. Then n^2 = 3*16+16 = 64 is a solution.