This is not information Google Analytics is set up to provide. As Stephen mentions in his comment, if you could inform GA of a visit being a user's first, via UTM parameters, that would work.
That seems like a difficult thing to guarantee, though. Without that, you may be able to get at a portion of the information you want, though with limitations.
One note: a session can last longer than 30 minutes. Sessions end after 30 minutes of inactivity or at midnight, but if someone was clicking around on your site consistently for an hour one afternoon, that would be recorded as a single session.
I am assuming here that by conversions you mean "official" GA conversions: goal or transaction.
In the Conversions > Multi-Channel Funnels > Model Comparison Tool report, make the following settings changes:
Conversion dropdown: your specific conversion type
Lookback Window: 90 days
"Last Interaction" model: change to "First Interaction"
Landing Page URL (available under Other > Acquisition)
Reading the table:
The date range you set for the report as a whole defines when the conversions occurred. The lookback window is relative to the conversion date and maxes out at 90 days. So if your homepage has 10 conversions attributed to it with the date range set to include only yesterday, it means that 10 conversions happened yesterday with the property that the earliest session the user had in the 90 days leading up to yesterday began on your homepage (whether the conversion occurred within that session or not).
Typically, setting very large date ranges is bad for accuracy, because in the reports that aren't under Multi-Channel Funnels, you'll get sampled data - numbers extrapolated from a subset of the data. However, these reports have different sampling rules, and you won't get sampled until your date range contains over a million conversions.
If the visitor arrived for the first time four months prior to converting, you won't get their true initial landing page, but you should get a good sense regardless, unless you have a very long average time to conversion. The Time Lag report, also in Multi-Channel Funnels, can give you a sense of whether that might be the case.
Without you being able to tell GA when a session is someone's first, I believe this is as good as it gets. It should give you trends and comparative value at least, and if your typical time to conversion is short, it will give you more than that.