This question hinges on the definition of “humanoid.” The Federation has its own definition, so let’s start with that one:
United Federation of Planets’ definition
According to Memory Alpha, the Federation uses a fairly specific definition of humanoid. I don’t have access to the exact wording of the Federation’s definition, so I’ll have to hope that Memory Alpha summarized it correctly in their article:
A humanoid is an intelligent, bilaterally symmetrical, bipedal lifeform in possession of an upright posture, two arms, two legs, hands, feet, one thorax, a neck, and a head with a brain located in it.
Now let’s look at a Borg drone. I included Memory Alpha articles wherever possible, but none of the links have surprising definitions:
- Intelligent: It’s hard to say, while they’re plugged into the Collective. But if you remove one from the Collective (e.g. Seven of Nine), they exhibit intelligence.
- Bilaterally symmetrical, bipedal: Yes.
- Lifeform: I’d say the Borg are alive. This is perhaps debatable.
- Upright posture, two arms, two legs, hands, feet, one thorax, a neck, and a head: Yes.
- Brain: Yes. Seven of Nine was able to function outside the Collective, so they clearly did not remove her brain. Also, Memory Alpha shows a picture of a Borg drone with its brain exposed, which I decline to include here.
Overall, it looks like a Borg drone qualifies as humanoid under this definition. A couple points are debatable for a drone who’s in the Collective, but a former drone detached from the Collective is clearly humanoid.
Twenty-first century English
I don’t know what common usage is in the 24th century, but in this century, the definition is not very restrictive:
Having human form or characteristics
– Merriam-Webster’s Collegiate Dictionary
In other words, if it looks kind of like a human, you can say it’s a humanoid. This applies to a Borg drone.
Whatever Q was saying, he was probably either correct or lying, since he presumably has enough knowledge to avoid being mistaken. He either categorizes the Borg as being humanoids by the Federation’s definition, or he was using a looser definition of the word.