Yes, but they are different tenses and have slightly different meanings.
1. "I got good news"
This sentence is in the PAST tense and it means "I received good news" or "I heard good news". The verb here is 'get' in the past simple form (got) , used with the meaning of 'obtained'. You might say, for example, "I got some good news yesterday. The bank wrote to me saying that they've approved my business loan."
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2. "I have got good news" ( usually contracted to "I've got good news").
This sentence is in the PRESENT tense. It means that you have the news now. The verb here is 'have', while the word 'got' does not serve any function at all in terms of meaning.
There are some not very well educated people who drop the 'have' in #2. For example, some people say "I got a problem" instead of "I've got a problem". You may come across this in informal speech, for example in songs and film dialogues, but you should not copy it.
I hope that helps.