Diophantine equations $ax^4+by^2=c$ in rational numbers

Your curve is a genus 1 curve, usually expressed as $$ y^2= -abx^4+bc$$ (just by multiplying everything by $b$ and changing $y$ by $by$).

The curve has local points everywhere, but since it looks like has no rational points, you can try to do a 2-descent. This can be done easily with the TwoCoverDescent algorithm as explained in

Nils Bruin and Michael Stoll. Two-cover descent on hyperelliptic curves. Math. Comp., 78:2347--2370, 2009.

It is not guaranteed to succeed, but in the special case you are interested $a=3$, $b=1$ and $c=8076$ it does and the answer is that it has no rational points.

All this has been implemented in MAGMA. You can use the following code

P:=PolynomialRing(Rationals()); H:=HyperellipticCurve((-3)*P.1^4+8076); Hk:=TwoCoverDescent(H);#Hk;

It answers 0, so there is no 2-cover with rational points, so the original curve has no rational points.

Multiplying by $x^2$ and denoting $X:=-abx^2$, $Y:=ab^2xy$, we get an elliptic curve: $$Y^2 = X^3-ab^2cX.$$ If it turns out that this curve has zero rank, then the number of solutions is finite and they are given by torsion points.

Luckily this is the case for the given values $a,b,c$, while the torsion points do not give a solution to the original equation.