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I saw in a grammar book mentioned something like this :

Liz can speak French very well.

Then I saw another example saying like this :

  1. She often eats durians

  2. My mother always wipes the floor.

With those examples, I got a little bit confuse. Why is the 1st example not saying "speaks" as it is referring to Liz which is only 1 person? Is it because it is referring to Proper Noun ?

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The the verb in the first sentence is not speak, but can speak. Can is a modal verb indicating ability, possibility, or permission. The exact modality can only be known with additional context.

Can (and may), when used as modal auxiliaries, are always used with the bare infinitive form of a verb, which in this case would indeed be speak.

Liz speaks French very well.

Liz can speak French very well[, but she did not want to embarrass her sister].

The subject being plural or a proper noun makes no difference in the form.

The rivers can swell high with snowmelt well into the summer.

Eli Manning can boast that he still has more Super Bowl MVP awards than his brother.

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It's the bare infinitive, speak, because the sentence uses the modal can. When we use a modal like can, should, might, etc., we don't conjugate the verb after it.

For example,

He rides a bicycle. / He can ride a bicycle.
She paints well. / She might paint well.
Pat opens the door. / Pat should open the door.

Reference: modal verbs

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