I think you may need some more information to find an expression for L (or K).
From what we have, we know MP1/MP2 = w/r, and that lets us say K = aL.
Now we look at our profit, the thing we want to maximize. Substituting K = aL into Q - TC, we get L(pb - 4w), where $b = pa^{3/4}$, and p is the price.
As long as p > 4r/3, we can increase profit forever by increasing L, as long as we increase K proportionally.
This assumes p is fixed, and labor and capital are infinitely available at r and w. So that's where our constraint has to come from. (Unless we're simply given a fixed value for Q, in which case we solve from K = aL.)
If we have a demand curve, and we're the only supplier, we increase L and K until Q becomes such that p = 4r/3, or we hire people until we can't get aL units of capital, or we increase K until we can't get K/a units of labor. If we're not the only supplier, we play a game with the other suppliers and fix Q ourselves.
But I don't think we can maximize profit without that extra information, because the functional form of Q together with the fact that L and K are linearly related means Q is linearly related to L or K, and the cost function is as well.