In the Catholic Church, Immaculate Conception (usually capitalized) refers to the conception of Jesus’ mother, not of Jesus. The belief holds that Mary, Jesus’ mother, was conceived without the stain of original sin, but nowhere does it say that Mary’s mother was a virgin or that Mary was conceived nonbiologically. Immaculate here means free from stain or pure.
Virgin birth is the term for Jesus’ birth. The story goes that God caused Jesus to appear miraculously with no natural father, and that Mary remained a virgin.
The problem with these terms is that virgin birth doesn’t cover the moment when Jesus was miraculously conceived. For this, miraculous conception might work, though perhaps there are established alternatives we don’t know about.
The use of Immaculate Conception in place of virgin birth is common. Here, for instance, the writers are clearly not referring to the conception of Mary:
[I]t has prompted many researchers to imagine a future when male attributes would have to find another way to hitch a ride into future generations: perhaps even leading to a new dawn of immaculate conception. [GreenAnswers]
The swami’s birth in Andhra Pradesh, which his mother claims was the result of an immaculate conception, is shrouded in mystery. [Telegraph]
She wants the child to be Italian and it’ll be an immaculate conception, since she wants to be artificially inseminated! [PopCrush]
And though virgin birth doesn’t always work in place of Immaculate Conception in cases like the above, it is occasionally useful—for example:
No one in that world knew, for example, that the virgin birth was a ninth-decade addition to the Christian story. [Brisbane Times]
[T]he doctrine of the Virgin Birth, that the Mother of God conceived “without carnal knowledge”, demeans all womanhood. [Irish Independent]
ObamaCare would emerge from the House by way of a virgin birth. [The Hill]