Are Finnish people and Turkish people both originated from Huns?

No. Genetic studies have revealed the following about the origins of the Finnish people:

Recently, the use of mitochondrial "mtDNA" (female lineage) and Y-chromosomal "Y-DNA" (male lineage) DNA-markers in tracing back the history of human populations has been started. For the paternal and maternal genetic lineages of Finnish people and other peoples, see, e.g., the National Geographic Genographic Project and the Suomi DNA-projekti. Haplogroup U5 is estimated to be the oldest mtDNA haplogroup in Europe and is found in the whole of Europe at a low frequency, but seems to be found in significantly higher levels among Finns, Estonians and the Sami people.[39] Of modern nationalities, Finns are closest to Cro-Magnons in terms of anthropological measurements.[40]

With regard to the Y-chromosome, the most common haplogroups of the Finns are N1c (58%), I (29%), R1a (7.5%) and R1b (3.5%).[41] Haplogroup N1c, which is found only in a few countries in Europe (Latvia, Lithuania, Estonia, Finland and Russia), is a subgroup of the haplogroup N (Y-DNA) distributed across northern Eurasia and estimated in a recent study to be 10,000–20,000 years old and suggested to have entered Europe about 12,000–14,000 years ago from Asia.[42]

Finns show very little if any Mediterranean and African genes but on the other hand almost 10% of Finnish genes seem to be shared with Siberian populations. Nevertheless more than 80% of Finnish genes are from a single ancient North-European population, while most Europeans are a mixture of 3 or more principal components.[43]

The bottom line is that Finns are descended from an ancient line of peoples from the far north of Asia who migrated into Scandinavia 10,000 or more years ago, long before Huns or Turks or existed. Due to fluctuations in glacial cover after the last Ice Age, they may have been isolated from other European population groups farther south for long periods, since they show little genetic input from more southern regions. Their language is NOT Turkic and is not linguistically related. And they are NOT descended from the Huns. They are far older than the Huns which had a central Asian origin slightly more than 2.000 years ago.