I have already read the following (this one and this one too) discussions on Stack Exchange and they have not answered my query. Proposition 2.2.14 asks the reader to prove that:
Proposition $2.2.14$ (Strong principle of induction). Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number $m$. Suppose that for each $m \ge m_0$, we have the following implication: if $P(m')$ is true for all natural numbers $m_0 \leq m' < m$, then $P(m)$ is also true. (In particular, this means that $P(m_0)$ is true since in this case, the hypothesis is vacuous .) Then we can conclude that $P(m)$ is true for all natural numbers $m\geq m_0.$ (Hint: define $Q(n)$ to be the property that $P(m)$ is true for all $m_0≤m<n;$ note that $Q(n)$ is vacuously true when $n<m_0.$)
Instead of what Tao suggests, can I let $Q(n)$ to be the property that $P(m)$ is true for all $m_0\le m<m_0+n$?
Yes. That would be saying exactly the same thing. Picking $m_0+n$ as the upper bound is the same as picking $n'=m_0+n$ as the upper bound like Tao suggested. Just a different way of writing it.