The answer is $\boxed{q=1/4,3/11,1/3.}$ Let $q=:\frac mn$ where $\gcd (m,n) =1.$The key is $$\frac mn=q=\frac{\text{time of the tortoise can see the hare}}{\text{all time}}=\frac{S(m,n)}{4mn},$$here define $$S(m,n):=\#\left\{0\le k\le 4mn-1:\lfloor k/m\rfloor\equiv\lfloor k/n\rfloor\pmod 4\right\}.$$It is not difficult to calculate $$S(m,n)=\begin{cases}mn& 2\mid mn\\mn+3&m\equiv n\in\{1,3\}\pmod 4\\mn+1&m\equiv -n\in\{1,3\}\pmod 4\end{cases}$$We finally get $q=1/4,3/11,1/3$ by substitution.$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\Box$