Answer: No.
Suppose that $N = m^4 + 2mn + n^2 - 2021$ is the product of three or more consecutive integers. Then $N$ and $m^4 - m^2 = m[(m-1)m(m+1)]$ are divisible by $3$. Thus $(m+n)^2 \equiv N - (m^4 - m^2) + 2021 \equiv 2 \pmod{3}$, which is a contradiction, since $2$ is not a quadratic residue modulo $3$.