  In this paper, we determine all the positive integers $a, b$ and $c$ such
that every nonnegative integer can be represented as $$
f^{a,b}_c(x,y,z,w)=ax^2+by^2+c(z^2+zw+w^2) \,\, \textrm{with}
\,\,x,y,z,w\in\mathbb{Z}. $$ Furthermore, we prove that $f^{a,b}_c$ can
represent all the nonnegative integers if it represents $n=1,2,3,5,6,10.$
