As the ice knife spell can do damage in two separate ways, with one defined to happen after the other, would a caster damaged by both need to make multiple concentration checks?
The target would need to make up to two saves.
The short answer:
Since Ice Knife deals damage twice, the target would indeed make up to two saves.
The long answer:
The rules for Concentration say (emphasis mine):
Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your Concentration. The DC equals 10 or half the damage you take, whichever number is higher. If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.
The examples provided in the rule above aren't directly applicable to this situation since both damage sources in Ice Knife are coming from the same spell, but the portion about making multiple saving throws is still pertinent.
We have additional relevant information in how and when we roll for damage. We roll damage as soon as we hit with an attack:
3. Resolve the attack. You make the attack roll. On a hit, you roll damage...
When we make the the initial attack roll, we roll for damage on a hit. Then our target must make a DEX save, which is part of a separate damaging event that happens after we've already tried to hit and deal damage once. Since we're attempting to deal damage a second time—even with the same spell—the target may need to make an additional concentration check.
An example for our argument is Eldritch Blast. From the description:
The spell creates more than one beam when you reach higher levels.... You can direct the beams at the same target or at different ones. Make a separate attack roll for each beam.
Let's say we cast Eldritch Blast when we're level 5, giving us two beams, and we choose the same target for each beam. Since we already know we roll for damage whenever we hit with an attack, and we know we're making multiple attack rolls, we therefore know we must make multiple saving throws to maintain our concentration if we're hit multiple times, since each beam in this case is its own source of damage.
Yes. Two saves may be required.
Two concentration saving throws are required as there are two discrete damage instances. In addition to being explicitly stated as separate and ordered damage in the ice knife description, "Hit or miss, the shard then explodes", they're also completely different mechanics.
I would argue 2 concentration checks are needed, since if one passes the dex saving throw they don't take the cold damage whatsoever. That being said, for this to happen the caster must hit the target, and the target must fail the dex save for 2 concentration checks to be required. The DC for each is relative to half the damage taken (for scaling purposes as the lvl increases), or DC 10, whichever is higher.