Step 1: Concept
Diborane ($B_{2}H_{6}$) contains two types of bonds: terminal two-center two-electron ($2c-2e$) bonds and bridging three-center two-electron ($3c-2e$) bonds.
Step 2: Meaning
A bridge bond involves three atoms ($B-H-B$) sharing a single pair of electrons, making it an electron-deficient banana bond configuration.
Step 3: Analysis
There are exactly two bridging $B-H-B$ bonds present in a diborane molecule. Since each three-center bridge bond contains $2$ electrons, the total number of electrons involved across both bridge bonds is:
$2 \text{ bridge bonds} \times 2 \text{ electrons per bond} = 4 \text{ electrons}$.
Step 4: Conclusion
Thus, the total count of electrons allocated exclusively within these bridging centers is 4.
Final Answer: (C)