The DirectSearch package should be downloaded from http://www.maplesoft.com/applications/view.aspx?SID=101333 and installed in your Maple >=12. In particular, that package includes the SolveEquations command. Let us try it.

DirectSearch:-SolveEquations(abs(a-b)+sqrt(2*b+c)+c^2-c+1/4 = 0, tolerances = 10^(-14), AllSolutions, solutions = 2);

The output suggests {a = -1/4, b = -1/4, c = 1/2} is the only solution of the equation under consideration. Using the assume=integer option, one can verify that guess.

DirectSearch:-SolveEquations(eval(abs(a-b)+sqrt(2*b+c)+c^2-c+1/4 = 0, [a = (1/4)*A, b = (1/4)*B, c = (1/4)*C]), {abs(A) <= 10^4, abs(B) <= 10^4, abs(C) <= 10^4}, tolerances = 10^(-14), AllSolutions, solutions = 2, assume = integer, evaluationlimit = 30000);