Yes…and No.
If you consider the Person Jesus, then in his full human nature He would feel emotions just as we do. However, if we are speaking of the God of Divinity, then it is quite different.
God is unchanging and unchangable. He is perfect and complete. While we can change another person by hurting them causing them to feel anger, pain, or sadness or other negative emotions, it is impossible to change God. God likewise has no body. He does not cry, blush, tremble, or sulk.
God is about relationship. Through our free will we make choices and these choices do not change God, but rather our relationship with Him. When we commit an offense against God’s will, when we have broken our relationship we say we have offended Him.
Think of “offending” not as hurting His feelings, but of COMMITTING AN OFFENSE.