I hope your belief rests on more evidence than that. Paul calls Timothy "my son" in several places, both in the first and third person, and not merely "our brother":
1 Corinthians 4:17 For this reason I have sent to you Timothy, my son whom I love, who is faithful in the Lord.
Philippians 2:2 But you know that Timothy has proved himself, because as a son with his father he has served with me in the work of the gospel.
1 Timothy 1:2 To Timothy my true son in the faith: Grace, mercy and peace from God the Father and Christ Jesus our Lord.
Peter and John also reference the other believers:
1 Peter 5:12 With the help of Silas, whom I regard as a faithful brother, I have written to you briefly, encouraging you and testifying that this is the true grace of God.
3 John 12 Demetrius is well spoken of by everyone—and even by the truth itself. We also speak well of him, and you know that our testimony is true.
Paul was called to the Gentiles, not the Jews, while Peter was called to the Jews (Galatians 2:7-9; Romans 11:13 and 15:16; Ephesians 3:8).
Nope, it is not the only evidence but that Hebrews ending is packed with enough information to make someone conclude that it was Paul who wrote it. Let's have a look again:
Heb 13:
23I want you to know that our
brother Timothy has been released. If he arrives soon, I will come with him to see you.
24Greet all your leaders and all the Lord’s people.
Those from Italy send you their greetings.
25Grace be with you all.
1. The word Italy alone disqualifies Peter and many other apostles because it
suggests that the letter was being written from Italy or by someone who is from Italy. He is sending regards from
those from Italy, meaning that he is in Italy himself or from Italy. If Peter was an apostle to the Jews, what would he be doing in Italy?
Paul, as much as he was an apostle to the gentiles, preached the gospel in Jerusalem and outside Jerusalem into other nations- we all know that.
2. Only Paul would be aware of the happenings in Italian prisons because he was also held there at one moment and surprise surprise, Philippians was written by Paul and Timothy in an Italian prison as suggested here:
Philippians 1:
1Paul and Timothy, servants of Christ Jesus,
To all God’s holy people in Christ Jesus at Philippi, together with the overseers and deacons
a :
2Grace and peace to you from God our Father and the Lord Jesus Christ......
7It is right for me to feel this way about all of you, since I have you in my heart and,
whether I am in chains or defending and confirming the gospel, all of you share in God’s grace with me. .....
12Now I want you to know, brothers and sisters,
b that what has happened to me has actually served to advance the gospel.
13As a result,
it has become clear throughout the whole palace guard c and to everyone else that I am in chains for Christ.
14A
nd because of my chains, most of the brothers and sisters have become confident in the Lord and dare all the more to proclaim the gospel without fear.