Mendeleev predicted the existence of two elements and named them eka-silicon (later called germanium) and eka-aluminium (later called gallium). How many valence electrons are present in each one of them?

The two newly predicted elements by Mendeleev were termed Eka-silicon and eka-aluminium. They were termed like this because of the similarity of their properties towards aluminium and silicon.

The Eka-silicon was later called germanium, and eka-aluminium was later known as gallium. 
1. Germanium (Ge) - Eka-silicon: 
Germanium (Ge) has an atomic number of 32. It is located in Group 14 of the periodic table. Elements in Group 14 have 4 valence electrons, and thus, germanium has a valency of 4.

2. Gallium (Ga) - Eka-aluminium: 
Gallium (Ga) has an atomic number of 31. It is located in Group 13 of the periodic table. Elements in Group 13 have 3 valence electrons, and thus, gallium has a valency of 3.

Hence, Germanium and gallium have $4$ and $3$ valence electrons, respectively.