which one is correct ?

這個要看情況囉。

on the coming Sunday 因為有 the，語氣上便「強調」了「是在接著的星期天」這件事。相反，另外那個沒有 the，所以語氣上便少了那個「強調」。

縱使語氣上有分別，但兩者意思是沒有分別的，並沒有對或錯，只在乎你語氣上是不是想「強調」「是在接著的星期天」這件事。

on the coming Sunday, this coming Sunday ~ correct

upcoming (adj, before a noun) = going to happen soon
美國用多, 英國用 forthcoming

upcoming event (election, concert), book, film, CD

用在時間, 只有 upcoming month, year, season ~ 美國用
少見 upcoming Sunday = 即將來臨星期日
是有 upcoming Sunday service (event)

七瀨 遙 ( 博士級 4 級 ) 「有 the，語氣上便「強調」了 ... 沒有 the，所以語氣上便少了那個「強調」。」

有THE是對， 冇THE 是錯，唔好亂咁寫教錯人。

On the coming Sunday

, as this phrase consist of the word "the", it gives the sentence more clarity that you are indicating that particular Sunday and emphasises it is only on that day, not any others. You can also rephrase it as : " the upcoming Sunday" , remember to put the word "the" to enhance clarity.

Native speakers would not or would hardly say " on coming sunday", because this does not directly indicate which particular upcoming Sunday you are talking about, also because using " the " is so common (in this situation ), saying the phrase " on coming Sunday" would seem a bit strange to native speakers of English.
Therefore, I do not recommend you to use the latter phrase:D

on the coming sunday

"on the coming Sunday"

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