Adam Thomas and Claudia Patatas gave their child the middle name Adolf, a court's heard.
The 22-year-old and 38-year-old from Waltham Gardens are on trial at Birmingham Crown Court, accused of being members of National Action - which they deny.
The prosecution alleged the name 'Adolf' was in honour of the infamous Nazi leader.
The Crown also claimed the couple were pictured at home with another man, a convicted racist and "vehement Nazi", who was holding a Swastika flag and performing a Hitler-style salute over their baby.
Co-defendant Daniel Bogunovic, 27, of Crown Hills Rise in Leicester, is also in the dock facing the same membership charge.
Opening the case, Barnaby Jameson, prosecuting, told jurors on Tuesday all three defendants were alleged members of National Action - a group "so extreme and violent" it was banned by the Government in December 2016.
He said the case was about a "terror spread by a small cell of fanatics", built on a racist ideology and warned jurors they would find much of the evidence "upsetting and disturbing".
The barrister added the case would detail the "hate crime committed by National Action members and the ruthlessness with which they were prepared to spread terror".
Mr Jameson said Bogunovic and Thomas had a "particular interest" in owning machetes, and that bomb-making instructions were found on the second man's computer.
Turning to Thomas and Patatas' child, he told jurors: "The Crown can tell you that one of his middle names was Adolf."
He added: "Given that the child was born almost a year after National Action was banned, you may think the use of the name 'Adolf' - even as a middle name - was of significance."
Mr Jameson said: "This case is about a specific type of terror.
"It is a terror fuelled by hatred and division. It is a terror born out of fanatical and tribal belief in white supremacy.
"It is a terror that regards anyone who falls outside a cult of violent white racial supremacy as sub-human.
"Those that fall into the sub-human category are primarily blacks, Jews and Asians.
"Others in that category include gays, communists and feminists.
"The cult had particular venom for female Labour MPs perceived as sympathetic to migrants.
"It is a terror that can be summarised in two words: 'White Jihad' - in other words, a white holy war.
"This terror was spread by a small cell of fanatics of whom you will hear a great deal."
He added the case would take jurors to "a world in which any right-thinking person would wish did not exist".
Mr Jameson said: "It is a world of banner-carrying paramilitaries aping Hitler's SS.
"It is a world of Swastikas and the burning crosses of the Ku Klux Klan, otherwise known as the KKK.
"It is the world of a group called National Action.
"It is the world of a group so extreme and violent the Government banned it at the end of 2016 under the terrorism legislation."
Jurors also heard that photos of an "arsenal" of weaponry which the "extremist's extremist" Vehvilainen had built up were found on Thomas and Patatas' phones.
Thomas also stands accused of a having a terrorist document, namely the Anarchist's Cookbook.
Patatas, Thomas and Bogunovic deny all the charges and the trial, expected to last four weeks, continues.