Can God Lie? Can God make a rock he can't move?
The root of the questions actually is questioning the attributes of Allah(SWT) we muslims believe in. It's the same as saying "Well, if attributes of Allah(SWT) give him unlimited power, can he change his attributes to a limited one? Can God be Human if he can do anything?"
To furthermore deny the attributes of Allah(SWT) they start making comparisons, trying to look for similiarities between them and One who has nothing similiar to Him by asking questions such as "We can make something we can't lift, why can't God?" The very question itself is a contradiction for how can one find similiarities to one that has no similiarities?
It's similiar to saying "one of the Attributes of Allah(SWT) has these attributes:
Al-'Awwal
The First, The One whose Existence is without a beginning.
Al-'Akhir
The Last, The One whose Existence is without an end.
Can he change them as he has the ability to do anything?"
These questions are nonsense to a Muslim because in order to answer the question the Muslim will have to change an Attribute of Allah(SWT) (which makes that Muslim a believer) into an attribues which makes him a desbeliever.
So instead of going round the bend, why don't just come right out and ask me whether I will deny the attributes of Allah(SWT) or even entertain the thought of Allah(SWT) being similiar because the question is no diffrent.
Immam Tahawi Says:
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"knowledge is of two types: the humanley accessible and humanley inaccesibile, To either deny accissible knowledge or to claim the inaccessible is disbelieve. Faith is not sound unless accessible knowledge is embraced and the persuit of Inccessible is abandoned."
Human imagination cannot consptualise Him, nor can human undertaking rasp His reality.
As he was in eternity attributed with qualities, so He remains forever described with the same attributes.
It is not after creating that He merits the name "The creator" [al- Khaliq], nor through originating His creatures that he merits the name "The Originator" [al-Baari]
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And
we know our limits and in inconclusive matters of knowledge, we assert,
"God knows best"
"O Allah! Verily, I ask you by my testifying that there is no God worthy of worship except You. You are the One, the Self-Sufficient Sustainer of all, Who does not give birth, nor were You born, and there is none comparable to Him.''