Luke 2:7 says:
7 And she brought forth her firstborn son, and wrapped him in swaddling clothes, and laid him in a manger; because there was no room for them in the inn.
This implies that they first looked for an inn (maybe a guest room). I can understand the significance of the Christ child being born into humble circumstances, but why would an expectant couple seek an inn?
And really, would an inn have given them more privacy (which the account clearly shows is something Joseph wanted for Mary)? Joseph and Mary must have known the prophecy that Jesus would be born in Bethlehem, so I doubt the birth was a surprise while they were there.
The only other instance of inn in the KJV New Testament is Luke 10:34, the parable of the good Samaritan:
34 And went to him, and bound up his wounds, pouring in oil and wine, and set him on his own beast, and brought him to an inn, and took care of him.
35 And on the morrow when he departed, he took out two pence, and gave them to the host, and said unto him, Take care of him; and whatsoever thou spendest more, when I come again, I will repay thee.
In this case, the host provided care to the person (in return for payment). If innkeepers sometimes provided care to their guests, was it common for women to give birth in inns?