Note that the difference between the coefficient of $x$ and the constant term begins at $-10$ and it ends at $10.$ Since it can only change in increments of $1$ at a time, it follows that at some point, the difference must have been $1.$ Since a polynomial of the form $$x^2+(k+1)x+k$$has roots $x=1$ and $x=k,$ it follows that the answer is yes, i.e. at some point, a quadratic with integer roots must have appeared on the board. $\square$