The magnetic flux linked with a coil is $\varphi$ and the EMF induced in it is  E, then  

Induced emf, $E=\frac{d\varphi }{dt}$ Therefore, $E=0when\frac{d\varphi }{dt}=0andviceversa.$

Even if $\varphi =0,\frac{d\varphi }{dt}$ may be non-zero. Similarly, if $\varphi \ne 0,\frac{d\varphi }{dt}$ my be zero. For example, if $\varphi ={\varphi }_0\sin \left(\omega t\right)\text{\hspace{0.17em}.}$ 

Then, $E=-\frac{d\varphi }{dt}=-{\varphi }_0\cos \left(\omega t\right)$