Yes, he does it all the time
The Bible is full of stories where God listens to humans and does what they ask.
More than once, when God was tempted to destroy all of Israel except Moses, Moses pointed out some reasons he shouldn’t do that. And God decided to let his people live. On the journey to the promised land, the Israelites said they wished they had meat, and God heard them and gave them meat.
When David asked God to stop killing innocent people instead of punishing the person who had displeased him, God answered his prayer. (He was only willing to do that after David made a sacrifice, though.) God didn’t ignore David; he listened to his cries for help.
When Solomon expressed his desire for the temple he built to be God’s home forever, God heard him and agreed to what he said. And just as Solomon asked, he promised to listen whenever people prayed toward that temple.
When God made Jeroboam’s hand shrivel up and Jeroboam asked a prophet to ask God to fix it, God did so. When Asa asked God to help him defeat the vast Cushite army, God struck them down. When Manasseh the evil king of Judah was captured by the Assyrians, he repented and prayed, and God let him go home. Hezekiah had Isaiah pray for God to listen to the Assyrians mocking him, and God responded by sending an angel to kill a bunch of Assyrians and scare the rest away.
When the Arameans tried to kill Jehoshaphat of Judah because they mistook him for the evil king of Israel, he cried out and God rescued him. Later, Jehoshaphat told God that a hopelessly vast army of multiple nations was attacking Judah. He pointed out that it was God’s fault that those nations were there, and that God was expected to save his people from disaster when they cried out to him at his temple. So he did.
Elisha prayed for God to bring a boy back to life, and to manipulate what certain people could or couldn’t see, and God did everything he asked. When Jeremiah spoke to God on behalf of other people, he turned his wrath away from them. Jonah was eaten alive by a fish, but then he called to God for help, and God listened and answered his cry.
God listens and complies even when you might think he wouldn’t
In the days of the judges, whenever Israel was taken over by their enemies, God would hear the people lamenting their oppression and would send someone to save them. Even after he claimed he was never going to save them again, he couldn’t help sending someone to rescue them when they asked him to.
The Israelites asked for a human king, and even though God was displeased, he listened to them and gave them a king. And when Samuel called on God to send a storm out of season as a sign of his disapproval, God did just what Samuel said. Then when the king needed to know why God had stopped helping him, he asked God to communicate through various instruments of divination to indicate who had done wrong, and God complied.
Abraham repeatedly asked God to spare the city of Sodom if he could find enough righteous people there, and God wasn’t bothered; he agreed every time. When God heard how upset Hezekiah was that God had decided he would never recover from his illness, God changed his plan and let Hezekiah recover. When Ezekiel pointed out how abhorrent God’s orders were, God changed his command to make it a little less horrible. When Amos objected to the things God was planning to do to his people, God canceled his plans.
When Gideon was skeptical and asked for miraculous proof that God was really speaking to him, God supplied the exact signs Gideon requested, twice. When Elijah wanted to show people that his God was a real God that would perform miracles on command in order to prove his existence, God cooperated and sent fire from heaven.
When Elijah wanted to murder a hundred men just because he could, God cooperated and sent more fire down from heaven. When some soldiers from Israel cried out to God to help them defeat their enemies, God helped them defeat their enemies. And then when some soldiers from Judah cried out to God to help them defeat Israel, God helped them do that, too. When Naaman asked to be allowed to bow down to an idol, God told him to go ahead.
This is a general rule
God said when his people called on him and prayed to him, he would listen to them. And he did. In fact, the Bible says God will be there for everyone who honestly calls on him. As long as you’re not too wicked, he will hear your prayers. For the righteous, prayer is powerful and effective. If you believe in Jesus, you can ask for anything in his name, and he will do it. Any time two Christians agree on what they’re asking for, God will do it for them. And if you believe you already have what you’re asking for, God will definitely answer your prayer.1
And even those simple requirements aren’t really required. Everyone who calls on the name of the Lord will be saved. God is good, therefore he will he will give good gifts to those who ask him, and he will satisfy everyone’s desires. You don’t have to be good yourself to get God to give you gifts; he gives generously to everyone without finding fault. If you don’t have what you want, it’s only because you haven’t asked God for it. Everyone who asks receives.
No (with one exception)
But the Bible also says God pays no attention when people plead with him to do something about all the wickedness they’re experiencing. It says he did listen to a human being once, when he granted Joshua’s request to make the day longer. But that was the only time. God had never listened to a human before, and he has never listened to a human since. So all the Bible’s claims about God listening to people are false.