Consider two particles (having same mass) moving with constant velocity very close to speed of light, say $0.99c$ in opposite directions. Will the magnitude of their relative velocities exceed the speed of light?
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No they will not exceed the speed of light. Because the relative speed formula for any particle with a rest mass goes:
$$v_\text{rel} = \frac{v_1 - v_2}{1 - \frac{v_1v_2}{c^2}}$$
If you use 0.99c and -0.99c you will see that their relative speeds wont exceed the speed of light.
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