Can we automate litigation hold on specific mailboxes and while onboarding new employees to a certain department

you can use an Azure Runbook:
https://www.howdoiuseacomputer.com/index.php/2021/09/13/azure-runbook-enable-exchange-online-litigation-hold/

Yea, you could craft this to do that:
https://learn.microsoft.com/en-us/microsoft-365/compliance/use-a-script-to-add-users-to-a-hold-in-ediscovery?view=o365-worldwide

or just create a group - put that group on a hold and ensure the new users are always added to it:

https://learn.microsoft.com/en-us/microsoft-365/compliance/create-ediscovery-holds?view=o365-worldwide

@Mohan Chinnaswamy

For Exchange online, there doesn't exist such function. Here may be a work around for you:

You could follow this blog to create a Task Scheduler to run script below daily, it will check and enable LitigationHold for mailbox.

#Conenct to Exchange online, remember to change the Accout and Password  
$Account = "******@domain.onmicrosoft.com"  
$PassWord = ConvertTo-SecureString -String "Password" -AsPlainText -Force  
$UserCredential = New-Object -TypeName "System.Management.Automation.PSCredential" -ArgumentList $Account, $PassWord  
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection  
Import-PSSession $Session -DisableNameChecking  
  
#Find mailbox which belong to Department 321, if LitigationHold isnot enabled, enable this function for that mailbox  
$users =  Get-User -Filter "Department -eq '321' -and RecipientType -eq 'UserMailbox'"  
ForEach($user in $users){  
    if (!(Get-Mailbox $user.name).LitigationHoldEnabled) {  
        Set-Mailbox $user.name -LitigationHoldEnabled $true -LitigationHoldDuration unlimited  
    }  
}  
  
#exit  
Remove-PSSession $Session  

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