Probably not, but there's enough gray area to allow it.
There are two requirements for the Unicorn Spirit's benefit to trigger:
- The druid must cast a non-cantrip spell
- That spell must restore HP to a creature
Goodberry is a level one spell, which satisfies the first condition but it does not restore HP to any creature. It creates berries.
HP are not restored by the spell's casting. That requires another creature to use their action and consume the berries. Another way of looking at it: if the druid casts Goodberry and no one consumes a berry, then condition number 2 above is not met.
That said, the the wording of the Unicorn Spirit reads "if" not "when." This nuance creates a gray area that could allow a DM could to rule that the combination works but the Unicorn Spirit's affect does not trigger unless/until a berry (probably just the first) is consumed.
After all, letting the first berry trigger the Spirit's affect would still be balanced compared to casting a different level one (or higher) healing spell. To allow it to trigger off every berry's consumption would definitely not be.