Timothy, do not be ASHAMED of the testimony about our Lord

Timothy likely experienced imprisonment (Heb 13:23), probably after Nero blamed the Christians for the fire in Rome. Paul tried to encourage him in 2Tm 1:

7 God gave us a spirit not of fear but of power and love and self-control. 8 Therefore do not be ashamed of the testimony about our Lord,

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Was Timothy ashamed of Jesus's testimony? Does this statement imply or infer that Timothy was?

No, at least not in the First-Order Logical sense of "imply" or "infer."

nor of me his prisoner,

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Was Timothy ashamed of Paul the prisoner?

Again, no.

but share in suffering for the gospel by the power of God,

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Expect suffering. It came with the job description.

11b I was appointed a preacher and apostle and teacher, 12 which is why I suffer as I do. But I am not ashamed, for I know whom I have believed, and I am convinced that he is able to guard until that day what has been entrusted to me.

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Look to Jesus.

16 May the Lord grant mercy to the household of Onesiphorus, for he often refreshed me and was not ashamed of my chains.

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Paul gave Timothy the example of Onesiphorus, who was not ashamed of Paul's imprisonment.

These two themes continued in 2:

3 Share in suffering as a good soldier of Christ Jesus.

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8 Remember Jesus Christ, risen from the dead, the offspring of David, as preached in my gospel, 9 for which I am suffering, bound with chains as a criminal.

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15 Do your best to present yourself to God as one approved, a worker who has no need to be ashamed, rightly handling the word of truth.

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Paul used the words "ashamed" 4 times and "suffer*" 4 times in 2Tm 1-2.

Was Timothy ashamed of Jesus's testimony?

Given Paul's repeated mention of suffering and shame, Timothy might have had some issues dealing with them at the time. Hopefully, after reading Paul's letter, Timothy's faith was reinforced.