In Indian English, it is widely used as "I have a doubt with .....". While in America English, it is used as "I suspect there is a problem with...".
What is the conceptual difference between these two? Which is recommended/correct way to use?
There is a proposed duplicate question that links to a lengthy previous discussion describing why "doubt" and "question" might be interchangeable in some cases and where this came from. If I understand the question correctly, and the American English tag, I don't think that thread really answers the question of the conceptual difference and proper usage in American English. I'll focus on that here.
"I have a doubt"
The relevant Merriam-Webster definitions of doubt. Start with the verb form:
To doubt isn't to question, it is to state a kind of "negativity".
The noun form (keep the verb form in mind for perspective on these definitions):
"I have a question"
"Question" is neutral, just a means to obtain information. The relevant Merriam-Webster definitions of question:
"I suspect there is a problem with..."
To "suspect" isn't to have a question, it is more to have an opinion. Merriam-Webster definitions of suspect:
There is a definition that is similar to doubt:
The relevant definition:
This is in someways the reverse of "doubt". Doubt is distrust, disbelief, or concern that something is wrong, and it may interfere with moving forward. It is similar to "worry" in that it isn't a very constructive mental state, perhaps other than to motivate you to do something about it.
If it does motivate you to do something, that might lead to an idea or discovery that can put you on the road to a solution. At that point, you might say, "I suspect there is a problem with..." That is a specific thing you believe, and it is constructive.
A place where "doubt" and "suspect" could be used to express similar meanings would be "I doubt X is true" vs. "I suspect X is false". Notice that the terms approach the issue from opposite perspectives, so they need to be stated in the reverse of each other.
These sentences have similar, but not identical meanings. "I doubt X is true" expresses a strong belief that it is not true. "I suspect X is false" is a suspicion but not a strong belief that it is the case. To use "doubt" with the same meaning as "suspect", you would need to add qualifiers to soften it, such as "I have some doubts about X being true". Or, you could strengthen "suspect" to make it more equivalent to "doubt", such as "I strongly suspect that X is false".
In a comment, Dan Bron wrote:
As an AmE speaker, I've always found the "doubt" construction (to mean "question") offputting. To my ear, it expresses a reluctance to admit you don't know something; rather, it seems to express "I know this, but there's room for some uncertainty, and I want to discuss that with you", as opposed to "you have information I don't have, and I want you to give it to me". It comes across as egotistical and circumlocutory. Same as when a phone operator in a call center asks "can you please confirm your address", implying they want to match it to their records, when really hey don't have it at all.
It depends on your need. If you're conversing with a non-native, say another Indian person, in a casual conversation, then using the former would be fine. It's more like an informal way of putting across your query. But in written communication, always go for the formal approach of the statement. "I have a query regarding this" is also good for written communication, instead of "I have a doubt".