I think it is a serious conversation which has become boring. Am I right?
There are multiple interpretations of this phrase depending on context. There is insufficient context in your question to say definitively one way or another.
In particular, two common, but distinct meanings of "heavy" lead to different interpretations.
First, "heavy" can mean weighty to indicate importance, significance, or great magnitude. In such a use, a "heavy conversation" is more likely to cause anxiety than boredom. (For example a heavy conversation between a lawyer and their client, or a doctor and their patient).
Second, "heavy" can stand for heavy going (ground so sodden that a horse makes slow progress) which would strongly indicate boredom.
In each case, "weighty conversation" or "the conversation was heavy going" would be clearer.
As the construction is both unusual and ambiguous, if the listener can be certain that it was used after some thought, it is likely that it is being used in contrast with the much more common "light conversation" meaning trivial or casual conversation -- small talk. "Heavy conversation" would be a construction used rhetorically as a deliberate contrast to such light conversation.