Deterministic integral of a martingale

Question

Let $Q_t$ be a martingale and $$ M_T := \int_0^T Q_t \, dt. $$ Then, is $M_T$ a martingale?

Answer 1

No. Take $Q_t = B_t$, where $(B_t)_{t\geqslant 0}$ is a standard Brownian motion. Then $d M_t = B_t dt$, so $M_t$ has drift.