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This sentence below is extracted from vocabulary for IELTS book (recording scripts part).

But nowadays all we seem to do is argue and that causes a lot of conflict between us.

Why she's saying argue? Shouldn't it be arguing or argument? And why conflict is not plural?

The verb "argue" is correct. Similar structures exist:

What she does is play tennis.

All I do is miss you.

What you have to do is be happy

The structure is subject-verb-complement. The complement is a bare infinitive (without "to"). Your example is more complex as the subject is "All we seem to do", but the bare infinitive is just "argue".

"Conflict" is not plural as it is used as a non-count noun: "How much conflict?"

The example is correct. "argue" here is a verb. In "All we seem to do is argue" "argue" is the action that they take. It could be reworded as "we are always arguing" where "are" is the verb, and "arguing" is a gerund.

In "that causes a lot of conflict between us" "conflict" is indicating a state, a continuing condition, rather than a distinct action. It could be reworded as "That causes many conflicts between us" referring to many distinct, individual conflicts. In this case the meaning is not much changed.

To use the noun "argument" in place of the verb "argue" in this sentence, the sentence would need to be rewritten to use "have arguments". For example:

But nowadays all we seem to do is have arguments and that causes a lot of conflict between us.

This is perfectly grammatical, but I think it is slightly weaker writing than the original form, because of the need for the auxiliary "have"

But nowadays all we seem to do is argue and that causes a lot of conflict between us.

There is an omission of to before the verb argue, which is common these days especially in American English informal language.

And to is optional in such sentences (i.e sentences in that form)

...is (to) argue...

... since the verb do has a role in permitting that; otherwise, that structure wouldn't work.

Also, notice that argue is a verb in its infinitive form and can't be alternated with a noun such as argument.

You may want to read this.