Fermat gave a smallest positive solution of {x + y = A^2, x^2 + y^2 = B^4},
(x,y)=(4565486027761, 1061652293520)([1].Dickson: p.620)

I show two solutions, one solution is same as Fermat's solution.

Solution

x + y = A^2.....................................................................(1)
x^2 + y^2 = B^4.................................................................(2)
First, we find a solution of equation (2).

Let x=m^2-n^2, y=2mn, B^2=m^2+n^2...............................................(3)

m=s^2-t^2, n=2st, B=s^2 + t^2...................................................(4)

Substitute (3) and (4) to (1), then (1) becomes to (5).

A^2 = t^4-4st^3-6s^2t^2+4s^3t+s^4...............................................(5)

Set X = t/s, then Y^2=X^4-4X^3-6X^2+4X+1........................................(6)

Transform (6) to Weierstrass form (7). 

V^2 = U^3-2U....................................................................(7)

By using Cremona's mwrank, we obtain the rank=1 and generator is P(-1,-1) on (7).

Hence, (7) has infinitely many rational solutions.

Small solutions are shown below.
      [X , Y,  A,  B]

2P   [-119, 120, 1, 28561]
3P   [2276953, -473304, 1343, 2325625] 
4P   [4565486027761, 1061652293520, 2372159, 4687298610289] 
5P   [-1223746274969130365039, 1319559556940881359600, 9788425919, 1799664515907016914961] 
6P   [54420629434406206268103685648441, -21864804036399372236043874332600,
      5705771236038721, 58648738806449243564537197828441]
7P   [214038981475081188634947041892245670988588201, 109945628264924023237017010068507003594693720,
      17999572487701067948161, 240625698472667313160415295005368384723483849]

Positive solutions are the case 4P and 7P in above solutions.
Solution of the case 4P is same as Fermat's solution.

I don't know whether there are infinitely many positive solutions.


Reference

[1].L.E. Dickson:History of theory of numbers, vol 2.