The answer is no.

If this is not true, let's assume that $(x - a)(x - b)(x - c) - 1 = p(x)q(x)$, where $p$ is linear and $q$ is quadratic. Then $$p(a)q(a) = p(b)q(b) = p(c)q(c) = -1$$

If the coefficients of $p$ and $q$ are integers, the value of these two polynomials can take only integer values. Therefore, in each product, one of the factor must be $1$ and the other one is $-1$. Hence either $p(x)$ takes the value $1$ twice or it takes the value $-1$ twice. However, a $1^{st}$ degree polynomial cannot take the same value twice. This is a contradiction.