Effect on T1 relaxation due to the application of the gradient In MRI, the T1 relaxation constant for a tissue type is defined as the time taken for the longitudinal relaxation to recover approximately to 63% of its initial value.

Is there any modification of the T1 time due to application of gradients. Suppose, after application of the RF pulse, the magnetization is tipped completely in the transverse place. If one waits, then 63% of the longitudinal magnetization (z-direction) will recover after T1 seconds. Now, say at the same time I apply a gradient along the x or y direction. Would this have any affect on the longitudinal relaxation time?
It does not have any practical effect on the $T_1$ relaxation, if there is a gradient present or not. 

Sure, from an academic point of view, there is an effect: Typically, in biomedical applications $T_1$ gets longer at higher field strengths. This is the reason why imaging sequences have to be adapted to the field strength of the machine, if $T_1$-dependent signal behavior is exploited (see Fluid Attenuation, or magnetization prearation in MPRAGE as an example). So, since you increase the magnetic field in $z$ diretion by a few mT, this would also influence $T_1$. However, I have never seen a practical application where this was a concern.