That would express the 13th of March 1999 as
xiij MART. MCMXCIX or
MART. xiij MCMXCIX
Convert the day
Convert the number of the date into Roman numerals using lower case letters. Romans had no lower case letters - they were an invention of medieval scribes and became formalised with the advent of printing in the 15th century. But as much latin was used in medieval times as in the ancient period. Often, they made numbers look prettier by using a 'j' instead of an 'i' in the final place in a number. Sometimes they would use a 'u' instead of a 'v' in a number. So you can decide if you want to make the number look a bit more special. Whatever looks best or right is acceptable - scribes went by rules of asthetics as much as arithmetic. Here is my choice. You - and your engraver - can make your own.
JANUARIUS, FEBRUARIUS, MARTIUS, APRILIS, MAIUS, JUNIUS, JULIUS, AUGUSTUS, SEPTEMBER, OCTOBER, NOVEMBER, DECEMBER.
or as they are more correctly written as Romans did not have a 'J' or a 'U' in their alphabet
IANVARIVS, FEBRVARIVS, MARTIVS, APRILIS, MAIVS, IVNIVS, IVLIVS, AVGVSTVS, SEPTEMBER, OCTOBER, NOVEMBER, DECEMBER.
Fortunately the Romans abbreviated them and you can safely use the first three letters followed by a stop. The only exceptions are MART. and SEPT. and only MART. looks odd. Few will notice if you omit the 'T'. And it doesn't matter too much if you choose JUL. rather than IVL. And you could use lower case letters if you wanted.
Converting the year
Look up the year you want here.
Then write the three of them down in order. In the UK and Europe we tend to write dates in the order day/month/year. In North America people write month/day/year. In Japan they use the more logical - and best for computers - year/month/day. Here are some examples.
xxij APR. MCMXLVIII
vij AVG. MCMLXXIX
JUN. xv MCMLXXV
IVL. xii MCMLXXVII