Statement-I: Boron easily forms B³⁺ ion.

Statement-II: All other elements of the 13^th group form only M⁺ ion due to inert pair effect.

As the Boron atom is small in size a large amount of energy is needed to remove 3 electrons from the boron atom. So Boron does not form B³⁺ ion. The atomic number of Boron is 5. Its electronic configuration is 1s ^[2]2s^[2]2p^[1].
When one electron is removed from the p orbital a He-like fulfilled s orbital is left. This is highly stable. So the second ionization enthalpy is quite high.
Again when one electron is removed a half filled orbital is left. So the third ionization enthalpy is also quite high.
Since the total energy needed to make B³⁺ is the total of all the ionization enthalpies an enormous amount of energy is required to form it. 

In Ga ,In and Tl, both +1 and +3 oxidation states are observed. The elements belong to 13 group.

For thallium +1 is more stable than +3 due to inert pair effect.