Can a character use his reaction in the first round of combat before their turn? There is no surprise involved here.
Yes, an unsurprised character may use their Reaction before their first turn
There are two relevant rules in the Player's Handbook, emphasis added for relevance:
Certain special abilities, spells, and situations allow you to take a special action called a reaction. [...]
When you take a reaction, you can't take another one until the start of your next turn. If the reaction interrupts another creature's turn, that creature can continue its turn right after the reaction.
—PHB, pg. 190
If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends. A member of a group can be surprised even if the other members aren't.
—PHB, pg. 189
So between these two sections, we glean that the only point where a Reaction is "refreshed" is at the start of a character's turn; there's no explicit point where the "initial Reaction" is granted, therefore it is available before the character gains its turn.
The only scenario where a character would not gain their Reaction is if they are surprised, or otherwise under an explicit effect that denies use of a character's Reaction.
A Character needs an action that allows them to use their Reaction
Most regular uses of Reactions are unaffected by occurring before the first turn a character takes (Attacks of Opportunities, Counterspell, etc.), but one common use is the Readied Action; per the rules, a character won't be able to react with a Ready Action, because they weren't able to set it up in the first place due to not having had their first turn yet.
Strictly RAW, "Actions in Combat" cannot be taken outside of Initiative order. Many DMs waive this restriction on a case-by-case basis because adjudicating this rule for every possible decision a character could make would be cumbersome, but as a general rule, allowing characters to Ready Action outside of combat, as a general rule, can lead to characters getting extra attacks, and is a concept that's meant to be encapsulated by Initiative order.
So unless your DM rules otherwise on the Ready Action, a character would not be able to use the Reaction granted by Ready Action before their first turn.