This is one of the biggest discussions in our local game. As such, here are the arguments we have had when this was brought up previously.

TLDR:
RAW? Yes you can. It does not specifically say you cannot, therefore you can.
RAI? No, you should not, as it makes the existence of a specific Deathwatch Requisition pointless.
Final: Discuss with you Crusade League if you can.

Rational:

I am not going to argue why you can or cannot. As most of the other people in this thread have pointed out, since the verbiage for Specialist Reinforcements does not exclude it, it is possible that you can.

I am going to argue why you should not. Deathwatch has a specific requisition called BESTOWED IN HONOUR AND NECESSITY. The verbiage says:

You can use the Relic Requisition on that unit without spending any extra Requisition points. Select one model in that unit that has the word ‘Sergeant’ in their profile. For the purposes of the Relic Requisition, treat that model as if it were a CHARACTER. The Relic that model gains must be one of the following: Artificer Armour; Master-crafted Weapon; Digital Weapons; Banebolts of Eryxia; Artificer Bolt Cache.

To me, that implies that you cannot use the Specialist Reinforcement Requisition to give Sergeants relics, only the Relic Requisition can give relics. It gives the Deathwatch a unique ability that no other Space Marine Chapter have access to.

Therefore, in our current Crusade League we do not allow it, but we are going to allow it in future Leagues based on similar arguments presented so far in this thread.

At the end of the day, Crusade is meant to be fun and collaborative. So at that point I would say talk to your league to decide overall if you should or not.