Answer
The electron configuration of $Ti^{2+}$ is $$[Ar]3d^{2}$$
$Ti^{2+}$ does not possess noble-gas configuration.
Work Step by Step
*RULES FOR WRITING ELECTRON CONFIGURATION FOR CATIONS:
1) Write the electron configuration of the neutral atom.
2) Find out the number of electrons to be removed from the neutral atom to create the cation.
3) Removing the electrons according to the following rules:
- The electrons occupying the subshells with the largest $n$ value will be removed first.
- If there are more than one subshell having the largest $n$ value, the electrons of the subshell with the highest $l$ value will be removed first.
1) The electron configuration of a neutral $Ti$ atom is $$[Ar]3d^24s^2$$
2) To make $Ti^{2+}$, 2 electrons need to be removed from a neutral $Ti$ atom.
3) Here we find that subshell $4s$ has the largest $n$ value $(n=4)$.
Subshell $4s$ is occupied by 2 electrons in a neutral $Ti$ atom. Therefore, to make $Ti^{2+}$, both of these 2 electrons would be removed.
So, the electron configuration of $Ti^{2+}$ is $$[Ar]3d^{2}$$
The above configuration is not the electron configuration of any noble gases. So $Ti^{2+}$ does not possess noble-gas configuration.