Nope.

1. The Qur'an doesn't contain any trace of homophobia and does not teach Muslims to be homophobic.

From the dictionary:
Homophobic means "having or showing a dislike of or prejudice against homosexual people." https://twitter.com/syafqz/status/1273221333237657603

2. The Qur'an rather goes against homosexuality. It tells the story of the people of Luth to tell about the evil act of homosexuality and its final consequences.

Homophobia/c does not equal homosexual/ity. Please don't conflate those two.

The Qur'an doesn't prohibit a thing and allow the other just for play and games. There are serious consequences on the table.

For example, alcohol was mentioned [2:219] to be of great sin and of benefit to the people. BUT the sin is greater than the benefit and hence prohibited.

We know how bad alcohol is long term and short term. It is an expensive drink to spoil the mind and break relationships.

Not to mention the issue of drink driving killing innocent lives, rupturing families.

Homosexuality as well, doesn't go without its consequences.

Contracting new diseases and health problems, breaking families, and goes against reproduction.

If can think of no benefit from this act. Zero, nothing, nil, null, tak ada, yillek, mei you.

3. For Muslims reading, yes the Qur'an mentions that the act is evil and corrupt. And yes people of Sodom was punished for their transgression.

But..

..they were rightly judged and retributed by their Creator and He has every right to do so. They ARE His creation. He owns them.

4. We (Muslims) on the other hand have no right to judge and retribute (except through the correct channel and with specific conditions i.e. Syari'ah).

We do not own them. They are not ours.

5. If by reading the Qur'an we feel obliged to judge others, then that's not the correct understanding.

If by reading the Qur'an we say that such evil people are hell-bound, then that's not correct either.

Those verses tell the stories of people of the fire not for us to pick and choose who goes into it, but rather as a reminder FOR US and them.

Yes for ourselves first.

Remember, on the day of judgement people will say "nafsi, nafsi" (myself, myself) and will not care for others.

On the day of judgement, Allah himself will be The Judge, The Most Just. He will be the One who showers people with Mercy and others with Retribution.

The job isn't ours. Our job is to convey and act according to the laws.

6. So, NO. The Qur'an does not teach homophobia. Rather, it contains teachings about good conduct and abstaining evil. And one of it is avoiding the act of homosexuality.

Wallahua'lam.

I should also add that if you read the Qur'an, you won't find anything like "You should hate the people of Lut or the homosexuals" or anything like it.

Read the exegesis (Tafsir) as well and do share if you find anything like it.