Let $N$ be the positive integer that you want to represent as $a^2+b^2+c^2+c$. By the three-squares theorem, $4N+1$ can be written as the sum of three squares, say $4N+1 = x^2+y^2+z^2$. Since all squares are $0$ or $1$ (mod $4$), exactly two of these three squares are even; without loss of generality, suppose $x$ and $y$ are even and $z$ is odd, and write $x=2a$, $y=2b$, and $z=2c+1$. Then $4N+1 = 4a^2 + 4b^2 + 4c^2+4c+1$, which means that $N=a^2+b^2+c^2+c$ as desired.
Of course, the three-squares theorem is not an easy one to prove, but it is standard and classical.