Are there times when I should not recite the Quran?

Yes, but only a few, and they depend on your situation rather than the time of day. There is no hour when reciting the Quran is forbidden. The things scholars ask you to avoid are reciting while bowing or prostrating in prayer, in the toilet, when you are very sleepy, and in ways that disturb others. Scholars also discuss reciting while in a state of major ritual impurity.

No forbidden hours

Voluntary prayer is not allowed at certain moments, such as sunrise and sunset, but reciting or reading the Quran is not restricted by the clock. You can recite after Fajr, after Asr, late at night or in the middle of the day. Before you begin, Allah tells us to "seek refuge with Allah from Satan, the accursed" (Quran 16:98).

Situations to avoid

In ruku' and sujud: the Prophet ﷺ said he had been forbidden to recite the Quran while bowing or prostrating. He told us to glorify Allah in bowing and to make earnest supplication in prostration (Sahih Muslim 479).
In the toilet or other unclean places: scholars say the Quran should not be recited in the bathroom. Finish before you go in and continue when you come out.
When very sleepy: if your tongue stumbles at night and you don't know what you are saying, the Prophet ﷺ advised going to sleep (Sahih Muslim 787).
When it disturbs others: the Prophet ﷺ once heard people reciting loudly in the mosque and told them not to raise their voices over each other (Sunan Abi Dawud 1332).
As mockery or entertainment: never recite it to joke or in a setting of sin.

When others are reciting

The Quran also says what to do when you hear it recited. If you can, stop talking and listen:

وَإِذَا قُرِئَ ٱلْقُرْءَانُ فَٱسْتَمِعُوا۟ لَهُۥ وَأَنصِتُوا۟ لَعَلَّكُمْ تُرْحَمُونَ "When the Quran is recited, listen to it attentively and be silent, so you may be shown mercy." Quran 7:204

Scholars understand this mainly for the prayer and the Friday sermon, though it is good manners elsewhere too.

Janabah and periods

The report of Ali says the Prophet ﷺ would recite the Quran in all conditions, except when he was junub, meaning in a state of major impurity after intimacy (Jami at-Tirmidhi 146). Most scholars therefore say to wash (ghusl) before reciting, though some scholars are more lenient. For menstruation, scholars have different views. See Can I recite the Quran from memory on my period?

Things that are fine

You can recite without wudu from memory, in bed, on your commute (keeping your attention safe) and from a phone. See Can I read the Quran in bed or lying down? Touching a printed copy is a separate question. See Can I read the Quran on my phone without wudu?

Key points

No time of day is forbidden for reciting the Quran.
Don't recite the Quran in ruku' or sujud, or in the toilet.
Rest if you are so sleepy you can't understand what you say.
Scholars differ on reciting in a state of major impurity; ask your imam.

Tip: If you can't recite because of one of these situations, don't feel you've missed out. Make dhikr and dua instead, then return to the Quran when you are able.