Is there any phrase, proverb or idiom that convey "the turtle quarreled with the lake"?

A similar English phrase is Don't bite the hand that feeds you, defined as:

to treat someone badly who has helped you in some way, often someone who has provided you with money.

More generally, it advises one not to treat badly ("bite") the entity that is systematically helping one ("the hand that feeds you").

While it is true that the conventional use of Don't bite the hand... means "Don't intentionally hurt the thing that's supporting you", there is some semantic leeway in bite, since biting can be intentional or accidental. This leeway allows for the proverb's extension into broader contexts where the subject accidentally hurts a thing that's supporting it (for example, a boy declaring independence from his mother, and thereby accidentally upsetting her).

I think the idiom Cut off your nose to spite your face (suggested elsewhere) is harder to extend into these broader contexts, suggesting as it does intentionally cutting off your nose with the malicious intent of hurting your face.

There is one that is similar, even if not quite identical, and that is "bite the hand that feeds you." The Free Dictionary describes it as: to severely criticize the person or organization that helps you or pays you.

For example:

Son: I do not need your money anymore, I don't want anything from you

Mom: pfff. You should not bite the hand that feeds you.

Probably not as good as the other options, but how about "Don't look a gift horse in the mouth" or "Don't tilt at windmills"? For background on the former, see http://idioms.thefreedictionary.com/look+a+gift+horse+in+the+mouth:

1. To be ungrateful to someone who gives you something; to treat someone who gives you a gift badly. (Usually with a negative.) McGraw-Hill Dictionary of American Idioms and Phrasal Verbs. © 2002 by The McGraw-Hill Companies, Inc.
2. To criticize or refuse to take something that has been offered to you. Etymology: based on the idea that you can discover a lot about a horse's condition by looking at its teeth. Cambridge Dictionary of American Idioms Copyright © Cambridge University Press 2003.
3. If someone tells you not to look a gift horse in the mouth, they mean that you should not criticize or feel doubt about something good that has been offered to you. Cambridge Idioms Dictionary, 2nd ed. Copyright © Cambridge University Press 2006. Reproduced with permission.
4. Be critical or suspicious of something received at no cost. ("Don't look a gift horse in the mouth," has been traced to the writings of the 4th-century cleric, St. Jerome, and has appeared in English since about 1500. It alludes to determining the age of a horse by looking at its teeth. The American Heritage® Dictionary of Idioms by Christine Ammer. Copyright © 2003, 1997 by The Christine Ammer 1992 Trust. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5. To be critical or suspicious of something one has received without expense. American Heritage® Dictionary of the English Language, Fifth Edition. Copyright © 2011 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.

Also consider this phrase:

"It is folly to live in Rome and fight with the Pope!"

This conveys the message somewhat.

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I can't think of an English cliche that fits perfectly, but such a person is too proud for charity. In order to accept the help they need, they would need to swallow their pride.

There are many English sayings related to denying how much you want something. In romance, you are playing hard to get, and in general, you are playing it cool. If you avoid a social event you might enjoy to signal that you're above it, you're too cool for school. In general, a person who makes decisions that promote a certain image of themselves, rather than decisions that fulfill their wants and needs, is living a lie.

Rosalind Fergusson, The Facts on File Dictionary of Proverbs (1983) has several relevant proverbs on the subject of pride:

Pride goes before, and shame follows after.

Pride and poverty are ill met, yet often seen together.

It is a proud horse that will not bear his own provender.

The mother in the OP's example might well consider any of these expressions an appropriate reply to her child, but I haven't heard any of them used in everyday English speech.

Fergusson also has this somewhat different proverb (which again I've never heard used in the wild):

The hog never looks up to him that threshes down the acorns.

The common thread here is ingratitude—although the hog certainly doesn't reject the acorns, as the child in the OP's post does the pocket money.

An expression that is used in situations where someone recklessly rejects a good offer or situation is

Sooner or later you'll come crawling back.

This expression most often arises when a person has rejected someone or something in the expectation of finding someone or something better.