We present the measurements related to the spin alignment of $K^{*}$ and $\phi$ vector mesons at mid-rapidity for Pb--Pb collisions at $\sqrt{s_{\rm nn}}$ = 2.76 TeV using the ALICE detector at the LHC. The second diagonal spin density matrix element ($\rho_{00}$) is measured from the angular distribution of the decay daughters of the vector meson in the decay rest frame, with respect to the event plane and the production plane. The $\rho_{00}$ values are found to be less than 1/3 at low $p_{\rm T}$ ($<$ 2 GeV/$c$) for both vector mesons. The observed deviations from 1/3 are maximal for mid-central collisions at a level of 3$\sigma$ for $K^{*}$ and 2$\sigma$ for $\phi$ mesons. As a control measurement the $\rho_{00}$ values were found to be consistent with 1/3, i.e., no significant spin alignment, for pp collisions with respect to the production plane. The results are qualitatively consistent with the expectation from a model that attributes the spin alignment to a polarization of quarks in the presence of large initial angular momentum and a subsequent hadronization by the process of recombination.