Often, but not always.
If the spell has variables based on the caster level, then those variables are also increased if your caster level increases (or decreases).
From the SRD:
A spell’s power often depends on its caster level, which for most spellcasting characters is equal to her class level in the class she’s using to cast the spell.
You can cast a spell at a lower caster level than normal, but the caster level you choose must be high enough for you to cast the spell in question, and all level-dependent features must be based on the same caster level.
In the event that a class feature or other special ability provides an adjustment to your caster level, that adjustment applies not only to effects based on caster level (such as range, duration, and damage dealt), but also to your caster level check to overcome your target’s spell resistance and to the caster level used in dispel checks (both the dispel check and the DC of the check).
So if your caster level for a spell is increased, everything that is based on caster level also increases, including range, duration and damage dice. The description of Spell Specialization is specific and does mention that all variables are increased, so there is no reason to think the damage shouldn't be increased aswell.
Treat your caster level as being two higher for all level-variable effects of the spell.
However, some spells have their damage dice fixed (like Flaming Sphere), and as such, their damage is not increased by your caster level.