This is a quote from a Japanese woman (the English translation is her own) saying that she would like to have a kid.
Finally I met my sweet sweet niece!!!!!!!! I...I...I want a baby, too lol
ついに、ついに姪っ子と初対面♪( ´θ｀)ノ 子宮がうずく（笑）
The Japanese wording was however somewhat unusual for me, as it literally means something along the lines that her womb is throbbing. Is this phrase commonly used, or was it just a unique way to say that she wants a baby?
I had to do some research to find out the answer to your question... Let's just, err, not discuss how I found this out.
「子宮がうずく」, according to the great internet at least, is an expression usually used under a sexual context meaning "to get really excited". And I mean, like.. really excited. I'll spare you the details, but it is generally used in situations like one where girls go to a bar and they see a really handsome guy, so they say that he's a ”子宮がうずく男". The nuance behind this is that regardless of whether or not they like that guy, they find him attractive, so you can kind of guess the rest. I am really not sure why that woman said that, though. I could only find sexual contexts for that expression. There is some chance she said it tongue-in-cheek, but I really can't say. I most certainly did not find any evidence pointing towards "wanting to have a baby".
Anyway this is what I could come up with, maybe a native speaker can tell you more.
When I was in middle age, I heard this phrase fairly often. 子宮が疼く- meaning a uterus gets excited refers to the sexual excitement of women, in bars and night clubs. It doesn’t apply to man’s sexual desire. Nor does it mean "I want to have a baby."
The phrase was used in the headlines of yellow papers and subtitles of erotic films called “Romance Porno”, and I thought the phrase is completely outdated now.
Definitely not common. In the US, I don't think people would use something like the direct translation "My uterus is itching" to express wanting a child.
To me, it has a more sexual connotation, to be sexually aroused, to want someone's baby. Whether or not the woman using this line meant that, I'm not sure, but I read it as "I want to go have someone's baby" or to be blunt, "I want to go have sex".
Either way, it's best not to use the line in public, and instead bed replaced with something more common like 「赤{あか}ちゃんが欲{ほ}しい」.
